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Section6.1Exponent Rules and Scientific Notation

Figure6.1.1Alternative Video Lesson

Subsection6.1.1Review of Exponent Rules for Products and Exponents

In Section 2.7, we introduced three basic rules involving products and exponents. We'll begin with a brief recap and explanation of these three exponent rules.

Product Rule

When multiplying two expressions that have the same base, simplify the product by adding the exponents.

\begin{equation*} x^m \cdot x^n = x^{m+n} \end{equation*}
Power to a Power Rule

When a base is raised to an exponent and that expression is raised to another exponent, multiply the exponents.

\begin{equation*} \left(x^m\right)^n = x^{m \cdot n} \end{equation*}
Product to a Power Rule

When a product is raised to an exponent, apply the exponent to each factor in the product.

\begin{equation*} \left(x\cdot y\right)^n = x^{n}\cdot y^{n} \end{equation*}
List6.1.2Summary of Exponent Rules
Checkpoint6.1.3

Subsection6.1.2Quotients and Exponents

Since division is a form of multiplication, it should seem natural that there are some exponent rules for division as well. Not only are there division rules, these rules for division and exponents are direct counterparts for some of the product rules for exponents.

Quotient of Powers

When we multiply the same base raised to powers, we end up adding the exponents, as in \(2^{2}\cdot2^{3}=2^{5}\) since \(4\cdot8=32\text{.}\) What happens when we divide the same base raised to powers?

Example6.1.4

Simplify \(\frac{x^5}{x^2}\) by first writing out what each power means.

Solution

Without knowing a rule for simplifying this quotient of powers, we can write the expressions without exponents and simplify.

\begin{align*} \frac{x^5}{x^2} \amp= \frac{x \cdot x \cdot x \cdot x \cdot x}{x \cdot x}\\ \amp= \frac{\cancel{x} \cdot \cancel{x} \cdot x \cdot x \cdot x}{\cancel{x} \cdot \cancel{x} \cdot 1}\\ \amp= \frac{x \cdot x \cdot x}{1}\\ \amp= x^3 \end{align*}

Notice that the difference of the exponents of the numerator and the denominator (\(5\) and \(2\text{,}\) respectively) is \(3\text{,}\) which is the exponent of the simplified expression.

When we divide as we've just done, we end up canceling factors from the numerator and denominator one-for-one. These common factors cancel to give us factors of \(1\text{.}\) The general rule for this is:

This rule says that when you're dividing two expressions that have the same base, you can simplify the quotient by subtracting the exponents. In Example 6.1.4, this means that we can directly compute \(\frac{x^5}{x^2}\text{:}\)

\begin{align*} \frac{x^5}{x^2} \amp= x^{5-2}\\ \amp=x^3 \end{align*}
Quotient to a Power

Another rule we have learned is the product to a power rule, which applies the outer exponent to each factor in the product inside the parentheses. We can use the rules of fractions to extend this property to a quotient raised to a power.

Example6.1.6

Let \(y\) be a real number, where \(y \neq 0\text{.}\) Find another way to write \(\left(\frac{7}{y}\right)^4\text{.}\)

Solution

Writing the expression without an exponent and then simplifying, we have:

\begin{align*} \left( \frac{7}{y} \right)^4 \amp= \left( \frac{7}{y} \right) \left( \frac{7}{y} \right) \left( \frac{7}{y} \right) \left( \frac{7}{y} \right)\\ \amp= \frac{7 \cdot 7 \cdot 7 \cdot 7}{y \cdot y \cdot y \cdot y}\\ \amp= \frac{7^4}{y^4}\\ \amp= \frac{2401}{y^4} \end{align*}

Similar to the product to a power rule, we essentially applied the outer exponent to the “factors” inside the parentheses—to factors of the numerator and factors of the denominator. The general rule is:

This rule says that when you raise a fraction to a power, you may separately raise the numerator and denominator to that power. In Example 6.1.6, this means that we can directly calculate \(\left( \frac{7}{y} \right)^4\text{:}\)

\begin{align*} \left( \frac{7}{y} \right)^4 \amp= \frac{7^4}{y^4}\\ \amp=\frac{2401}{y^4} \end{align*}
Exercises

Try these exercises that use the quotient rules for exponents.

Checkpoint6.1.8

Subsection6.1.3The Zero Exponent

So far, we have been working with exponents that are natural numbers (\(1, 2, 3, \ldots\)). By the end of this chapter, we will expand our understanding to include exponents that are any integer, including \(0\) and negative numbers. As a first step, we will focus on understanding how \(0\) should behave as an exponent by considering the pattern of decreasing powers of \(2\) below.

power product value
\(2^4\) \(=\) \(2 \cdot 2 \cdot 2 \cdot 2\) \(=\) \(16\) (divide by \(2\))
\(2^3\) \(=\) \(2 \cdot 2 \cdot 2\) \(=\) \(8\) (divide by \(2\))
\(2^2\) \(=\) \(2 \cdot 2\) \(=\) \(4\) (divide by \(2\))
\(2^1\) \(=\) \(2\) \(=\) \(2\) (divide by \(2\))
\(2^0\) \(=\) \(\mathord{?}\) \(=\) \(\mathord{?}\)
Table6.1.9Descending Powers of \(2\)

As we move down from one row to the row below it, we reduce the power by \(1\) and we remove a factor of \(2\text{.}\) The question then becomes, “What happens when you remove the only remaining factor of \(2\text{,}\) when you have no factors of 2?” We can see that “removing a factor of \(2\)” really means that we're dividing the value by \(2\text{.}\) Following that pattern, we can see that moving from \(2^1\) to \(2^0\) means that we need to divide the value \(2\) by \(2\text{.}\) Since \(2\div 2 = 1\text{,}\) we have:

\begin{equation*} 2^0 = 1 \end{equation*}

We exclude the case where \(a=0\) from this rule, because our reasoning for this rule with the table had us dividing by the base. And we cannot divide by \(0\text{.}\)

Example6.1.11

Simplify the following expressions. Assume all variables represent non-zero real numbers.

  1. \(\left(173 x^4 y^{251}\right)^0\)

  2. \((-8)^0\)

  3. \(-8^0\)

  4. \(3x^0\)

Solution

To simplify any of these expressions, it is critical that we remember an exponent only applies to what it is touching or immediately next to.

  1. In the expression \(\left(173 x^4 y^{251}\right)^0\text{,}\) the exponent \(0\) applies to everything inside the parentheses.

    \begin{equation*} \left(173 x^4 y^{251}\right)^0 = 1 \end{equation*}
  2. In the expression \((-8)^0\) the exponent applies to everything inside the parentheses, \(-8\text{.}\)

    \begin{equation*} (-8)^0 = 1 \end{equation*}
  3. In contrast to the previous example, the exponent only applies to the \(8\text{.}\) The exponent has a higher priority than negation in the order of operations. We should consider that \(-8^0 = -\mathopen{}\left(8^0\right)\mathclose{}\text{,}\) and so:

    \begin{align*} -8^0 \amp= -\mathopen{}\left(8^0\right)\mathclose{} \\ \amp= -1 \end{align*}
  4. In the expression \(3x^0\text{,}\) the exponent \(0\) only applies to the \(x\text{:}\)

    \begin{align*} 3x^0 \amp= 3\cdot x^0 \\ \amp= 3\cdot 1 \\ \amp= 3 \end{align*}

Subsection6.1.4Negative Exponents

In Section 2.7, we developed rules for simplifying expressions with whole number exponents, like \(0\text{,}\) \(1\text{,}\) \(2\text{,}\) \(3\text{,}\) etc. It turns out that these same rules apply even if the exponent is a negative integer, like \(-1\text{,}\) \(-2\text{,}\) \(-3\text{,}\) etc.

To consider the effects of negative integer exponents, let's extend the pattern we examined in Table 6.1.9. In that table, each time we move down a row, we reduce the power by \(1\) and we divide the value by \(2\text{.}\) We can continue this pattern in the power and value columns, paying particular attention to the values for negative exponents:

Power Value
\(2^3\) \(8\) (divide by \(2\))
\(2^2\) \(4\) (divide by \(2\))
\(2^1\) \(2\) (divide by \(2\))
\(2^0\) \(1\) (divide by \(2\))
\(2^{-1}\) \(\sfrac{1}{2}=\sfrac{1}{2^1}\) (divide by \(2\))
\(2^{-2}\) \(\sfrac{1}{4}=\sfrac{1}{2^2}\) (divide by \(2\))
\(2^{-3}\) \(\sfrac{1}{8}=\sfrac{1}{2^3}\)

Note that the choice of base \(2\) was arbitrary, and this pattern works for all bases except \(0\text{,}\) since we cannot divide by \(0\) in moving from one row to the next.

Note that if we take reciprocals of both sides, we have another helpful fact:

\begin{equation*} \frac{1}{a^{-n}}=a^n\text{.} \end{equation*}

Taken together, these facts tell us that a negative exponent power in the numerator belongs in the denominator (with a positive exponent) and a negative exponent power in the denominator belongs in the numerator (with a positive exponent). In other words, you can see a negative exponent as telling you to move things in and out of the numerator and denominator of an expression.

Remark6.1.13

Traditionally, when mathematicians simplify an expression they write the expression using only positive exponents. (Negative exponents aren't quite as “simple” as positive exponents.) This can always be accomplished using the negative exponent rule, and you will often be asked to state your final result using only positive exponents.

Try these exercises that involve negative exponents.

Checkpoint6.1.14

Subsection6.1.5Summary of Exponent Rules

Now that we have some new exponent rules beyond those from Section 2.7, let's summarize.

If \(a\) and \(b\) are real numbers, and \(n\) and \(m\) are integers, then we have the following rules:

Product Rule

\(a^{n} \cdot a^{m} = a^{n+m}\)

Power to a Power Rule

\((a^{n})^{m} = a^{n\cdot m}\)

Product to a Power Rule

\((ab)^{n} = a^{n} \cdot b^{n}\)

Quotient Rule

\(\dfrac{a^{n}}{a^{m}} = a^{n-m}\text{,}\) as long as \(a \neq 0\)

Quotient to a Power Rule

\(\left( \dfrac{a}{b} \right)^{n} = \dfrac{a^{n}}{b^{n}}\text{,}\) as long as \(b \neq 0\)

Zero Exponent Rule

\(a^{0} = 1\) for \(a\neq0\)

Negative Exponent Rule

\(a^{-n} = \frac{1}{a^n}\)

Negative Exponent Reciprocal Rule

\(\frac{1}{a^{-n}} = a^n\)

List6.1.15Summary of the Rules of Exponents for Multiplication and Division
Remark6.1.16Why we have “\(a \neq 0\)” and “\(b \neq 0\)” for some rules

Whenever we're working with division, we have to be careful to make sure the rules we state don't ever imply that we might be dividing by zero. Dividing by zero leads us to expressions that have no meaning. For example, both \(\frac{9}{0}\) and \(\frac{0}{0}\) are undefined, meaning no one has defined what it means to divide a number by \(0\text{.}\) Also, we established that \(a^0=1\) using repeated division by \(a\) in table rows, so that reasoning doesn't work if \(a=0\text{.}\)

Warning6.1.17A Common Mistake

It may be tempting to apply the rules of exponents to expressions containing addition or subtraction. However, none of the rules of exponents 6.1.15 involve addition or subtraction in the initial expression. Because whole number exponents mean repeated multiplication, not repeated addition or subtraction, trying to apply exponent rules in situations that do not use multiplication simply doesn't work.

Can we say something like \(a^n + a^m = a^{n+m}\text{?}\) How would that work out when \(a=2\text{?}\)

\begin{align*} 2^3 + 2^4 \amp\stackrel{?}{=} 2^{3+4}\\ 8 + 16 \amp\stackrel{?}{=} 2^{7}\\ 24 \amp\neq 128 \end{align*}

As we can see, that's not even close. This attempt at a “sum rule” falls apart. In fact, without knowing values for \(a\text{,}\) \(n\text{,}\) and \(m\text{,}\) there's no way to simplify the expression \(a^n + a^m\text{.}\)

Checkpoint6.1.18

As we mentioned before, many situations we'll come across will require us to use more than one exponent rule. In these situations, we'll have to decide which rule to use first. There are often different, correct approaches we could take. But if we rely on order of operations, we will have a straightforward approach to simplify the expression correctly. To bring it all together, try these exercises.

Checkpoint6.1.19

Subsection6.1.6Scientific Notation

Having just learned more about exponents, including negative exponents, we can discuss a format used for very large and very small numbers called scientific notation.

Subsubsection6.1.6.1The Basics of Scientific Notation

An October 3, 2016 CBS News headline 1 http://www.cnsnews.com/news/article/terence-p-jeffrey/federal-debt-fy-2016-jumped-142282704745246 read:

Federal Debt in FY 2016 Jumped \(\$1{,}422{,}827{,}047{,}452.46\)—that's \(\$12{,}036\) Per Household.

The article also later states:

By the close of business on Sept. 30, 2016, the last day of fiscal 2016, it had climbed to \(\$19{,}573{,}444{,}713{,}936.79\text{.}\)

When presented in this format, trying to comprehend the value of these numbers can be overwhelming. More commonly, such numbers would be presented in a descriptive manner:

  • The federal debt climbed by \(1.42\) trillion dollars in 2016.

  • The federal debt was \(19.6\) trillion dollars at the close of business on Sept. 30, 2016.

Unless we're presented with such news items, most of us deal with numbers no larger than the thousands in our daily life. In science, government, business, and many other disciplines, it's not uncommon to deal with much larger numbers. When numbers get this large, it can be hard to distinguish between a number that has nine or twelve digits. On the other hand, we have descriptive language that allows us grasp the value and not be lost in the sheer size of the number.

We have descriptive language for all numbers, based on the place value of the different digits: ones, tens, thousands, ten thousands, etc. We tend to rely upon this language more when we start dealing with larger numbers. Here's a chart for some of the most common numbers we see and use in the world around us:

Number US English Name Power of \(10\)
\(1\) one \(10^{0}\)
\(10\) ten \(10^{1}\)
\(100\) hundred \(10^{2}\)
\(1{,}000\) one thousand \(10^{3}\)
\(10{,}000\) ten thousand \(10^{4}\)
\(100{,}000\) one hundred thousand \(10^{5}\)
\(1{,}000{,}000\) one million \(10^{6}\)
\(1{,}000{,}000{,}000\) one billion \(10^{9}\)
Table6.1.20Whole Number Powers of \(10\)

Each number above has a corresponding power of ten and this power of ten will be important as we start to work with the content in this section.

This descriptive language also covers even larger numbers: trillion, quadrillion, quintillion, sextillion, septillion, and so on. There's also corresponding language to describe very small numbers, such as thousandth, millionth, billionth, trillionth, etc.

Through centuries of scientific progress, humanity became increasingly aware of very large numbers and very small measurements. As one example, the star that is nearest to our sun is Proxima Centauri 2 imagine.gsfc.nasa.gov/features/cosmic/nearest_star_info.html. Proxima Centauri is about \(25{,}000{,}000{,}000{,}000\) miles from our sun. Again, many will find the descriptive language easier to digest: Proxima Centauri is about \(25\) trillion miles from our sun.

To make computations involving such numbers more manageable, a standardized notation called scientific notation was established. The foundation of scientific notation is the fact that multiplying or dividing by a power of \(10\) will move the decimal point of a number so many places to the right or left, respectively.

Checkpoint6.1.21

Multiplying a number by \(10^n\) where \(n\) is a positive integer had the effect of moving the decimal point \(n\) places to the right.

Every number can be written as a product of a number between \(1\) and \(10\) and a power of \(10\text{.}\) For example, \(650 = 6.5 \times 100\text{.}\) Since \(100 = 10^2\text{,}\) we can also write

\begin{equation*} 650 = 6.5 \times 10^{2} \end{equation*}

and this is our first example of writing a number in scientific notation.

Definition6.1.22

A positive number is written in scientific notation when it has the form \(a \times 10^n\) where \(n\) is an integer and \(1 \le a \lt 10 \text{.}\) In other words, \(a\) has precisely one digit to the left of the decimal place. The exponent \(n\) used here is called the number's order of magnitude. The number \(a\) is sometimes called the significand or the mantissa.

Subsubsection6.1.6.2Scientific Notation for Large Numbers

To write a numbers larger than \(10\) in scientific notation, we write a decimal point after the first non-zero digit of the number and then count the number of places between where the decimal point originally was and where it now is. Scientific notation communicates the size of a number and the order of magnitude just as quickly, but with no need to write long strings of zeros or to try to decipher the language of quintillions, sextillions, etc.

Example6.1.23

To get a sense of how scientific notation works, let's consider familiar lengths of time converted to seconds.

Length of Time Length in Seconds Scientific Notation
one second 1 second \(1 \times 10^{0}\) second
one minute 60 seconds \(6 \times 10^{1}\) seconds
one hour 3600 seconds \(3.6 \times 10^{3}\) seconds
one month 2,628,000 seconds \(2.628 \times 10^{6}\) seconds
ten years 315,400,000 seconds \(3.154 \times 10^{8}\) seconds
79 years (about a lifetime) 2,491,000,000 seconds \(2.491 \times 10^{9}\) seconds
Checkpoint6.1.24
Checkpoint6.1.25

Subsubsection6.1.6.3Scientific Notation for Small Numbers

Scientific notation can also be useful when working with numbers smaller than \(1\text{.}\) As we saw in Table 6.1.20, we can denote thousands, millions, billions, trillions, etc., with positive integer exponents on \(10\text{.}\) We can similarly denote numbers smaller than \(1\) (which are written as tenths, hundreds, thousandths, millionths, billionths, trillionths, etc.), with negative integer exponents on \(10\text{.}\) This relationship is outlined in Table 6.1.26.

Number US English Name Power of \(10\)
\(1\) one \(10^{0}\)
\(0.1\) one tenth \(\frac{1}{10}=10^{-1}\)
\(0.01\) one hundredth \(\frac{1}{100}=10^{-2}\)
\(0.001\) one thousandth \(\frac{1}{1{,}000}=10^{-3}\)
\(0.0001\) one ten thousandth \(\frac{1}{10{,}000}=10^{-4}\)
\(0.00001\) one hundred thousandth \(\frac{1}{100{,}000}=10^{-5}\)
\(0.000001\) one millionth \(\frac{1}{1{,}000{,}000}=10^{-6}\)
\(0.000000001\) one billionth \(\frac{1}{1{,}000{,}000{,}000}=10^{-9}\)
Table6.1.26Negative Integer Powers of \(10\)

To see how this works with a digit other than \(1\text{,}\) let's look at \(0.05\text{.}\) When we state \(0.05\) as a number, we say “5 hundredths.” Thus \(0.05=5\times \frac{1}{100}\text{.}\) The fraction \(\frac{1}{100}\) can be written as \(\frac{1}{10^2}\text{,}\) which we know is equivalent to \(10^{-2}\text{.}\) Using negative exponents, we can then rewrite \(0.05\) as \(5\times10^{-2}\text{.}\) This is the scientific notation for \(0.05\text{.}\)

In practice, we won't generally do that much computation. To write a small number in scientific notation we start as we did before and place the decimal point behind the first non-zero digit. We then count the number of decimal places between where the decimal had originally been and where it now is. Keep in mind that negative powers of ten are used to help represent very small numbers (smaller than \(1\)) and positive powers of ten are used to represent very large numbers (larger than \(1\)). So to convert \(0.05\) to scientific notation, we have:

\begin{equation*} 0\overbrace{.\highlight{05}}^{2\text{ places}}=5\times 10^{-2} \end{equation*}
Example6.1.27

In quantum mechanics, there is an important value called the Planck Constant 3 en.wikipedia.org/wiki/Planck_constant. Written as a decimal, the value of the Planck constant (rounded to 4 significant digits) is

\begin{equation*} 0.0000000000000000000000000000000006626\text{.} \end{equation*}

In scientific notation, this number will be \(6.626\times 10^{\mathord{?}}\text{.}\) To determine the exponent, we need to count the number of places from where the decimal is when the number is written as

\begin{equation*} 0.0000000000000000000000000000000006626 \end{equation*}

to where it will be when written in scientific notation:

\begin{equation*} 0\overbrace{.\highlight{0000000000000000000000000000000006}}^{34\text{ places}}626 \end{equation*}

As a result, in scientific notation, the Planck Constant value is \(6.626 \times 10^{-34}\text{.}\) It will be much easier to use \(6.626 \times 10^{-34}\) in a calculation, and an added benefit is that scientific notation quickly communicates both the value and the order of magnitude of the Planck constant.

Checkpoint6.1.28
Checkpoint6.1.29
Checkpoint6.1.30

Subsubsection6.1.6.4Multiplying and Dividing Using Scientific Notation

One main reason for having scientific notation is to make calculations involving immensely large or small numbers easier to perform. By having the order of magnitude separated out in scientific notation, we can separate any calculation into two components.

Example6.1.31

On Sept. 30th, 2016, the US federal debt was about \(\$19{,}600{,}000{,}000{,}000\) and the US population was about \(323{,}000{,}000\text{.}\) What was the average debt per person that day?

  1. Calculate the answer using the numbers provided, which are not in scientific notation.

  2. First, confirm that the given values in scientific notation are \(1.96 \times 10^{13}\) and \(3.23 \times 10^8\text{.}\) Then calculate the answer using scientific notation.

Solution

We've been asked to answer the same question, but to perform the calculation using two different approaches. In both cases, we'll need to divide the debt by the population.

  1. We may need to be working a calculator to handle such large numbers and we have to be careful that we type the correct number of 0s.

    \begin{gather*} \frac{19600000000000}{323000000}\approx 60681.11 \end{gather*}
  2. To perform this calculation using scientific notation, our work would begin by setting up the quotient \(\frac{1.96 \times 10^{13}}{3.23 \times 10^8}\text{.}\) Dividing this quotient follows the same process we did with variable expressions of the same format, such as \(\frac{1.96 w^{13}}{3.23 w^8}\text{.}\) In both situations, we'll divide the coefficients and then use exponent rules to simplify the powers.

    \begin{align*} \frac{1.96 \times 10^{13}}{3.23 \times 10^8} \amp= \frac{1.96 }{3.23} \times\frac{10^{13}}{ 10^8} \\ \amp\approx 0.6068111 \times 10^5 \\ \amp\approx 60681.11 \end{align*}

The federal debt per capita in the US on September 30th, 2016 was about \(\$60{,}681.11\) per person. Both calculations give us the same answer, but the calculation relying upon scientific notation has less room for error and allows us to perform the calculation as two smaller steps.

Whenever we multiply or divide numbers that are written in scientific notation, we must separate the calculation for the coefficients from the calculation for the powers of ten, just as we simplified earlier expressions using variables and the exponent rules.

Example6.1.32
  1. Multiply \(\left( 2\times 10^5 \right)\left( 3\times10^4 \right)\text{.}\)

  2. Divide \(\dfrac{8\times 10^{17}}{4\times 10^2}\text{.}\)

Solution

We will simplify the significand/mantissa parts as one step and then simplify the powers of \(10\) as a separate step.

  1. \begin{align*} \left( 2\times 10^5 \right)\left( 3\times10^4 \right) \amp= \left( 2\times 3 \right)\times \left(10^5 \times 10^4 \right) \\ \amp= 6 \times 10^{9} \end{align*}
  2. \begin{align*} \frac{8 \times 10^{17}}{4\times 10^2} \amp= \frac{8}{4} \times \frac{10^{17}}{10^2} \\ \amp= 2 \times 10^{15} \end{align*}

Often when we multiply or divide numbers in scientific notation, the resulting value will not be in scientific notation. Suppose we were multiplying \(\left( 9.3\times 10^{17} \right)\left( 8.2 \times 10^{-6} \right)\) and need to state our answer using scientific notation. We would start as we have previously:

\begin{align*} \left( 9.3\times 10^{17} \right)\left( 8.2 \times 10^{-6} \right) \amp=\left( 9.3\times 8.2 \right)\times \left( 10^{17} \times 10^{-6} \right)\\ \amp= 76.26 \times 10^{11} \\ \end{align*}

While this is a correct value, it is not written using scientific notation. One way to covert this answer into scientific notation is to turn just the coefficient into scientific notation and momentarily ignore the power of ten:

\begin{align*} \amp=\highlight{76.26} \times 10^{11} \\ \amp= \highlight{7.626 \times 10^1} \times 10^{11} \end{align*}

Now that the coefficient fits into the proper format, we can combine the powers of ten and have our answer written using scientific notation.

\begin{align*} \amp=7.626 \times \highlight{10^1 \times 10^{11}} \\ \amp= 7.626 \times 10^{12} \end{align*}
Example6.1.33

Multiply or divide as indicated. Write your answer using scientific notation.

  1. \(\left( 8 \times 10^{21} \right)\left( 2 \times 10^{-7} \right)\)
  2. \(\dfrac{ 2 \times 10^{-6} }{ 8 \times 10^{-19} }\)
Solution

Again, we'll separate out the work for the significand/mantissa from the work for the powers of ten. If the resulting coefficient is not between \(1\) and \(10\text{,}\) we'll need to adjust that coefficient to put it into scientific notation.

  1. \begin{align*} \left( 8 \times 10^{21} \right)\left( 2 \times 10^{-7} \right) \amp= \left( 8 \times 2 \right)\times\left( 10^{21} \times 10^{-7} \right) \\ \amp= \highlight{16} \times 10^{14} \\ \amp= \highlight{1.6\times 10^1} \times 10^{14} \\ \amp= 1.6 \times 10^{15} \end{align*}

    We need to remember to apply the product rule for exponents to the powers of ten.

  2. \begin{align*} \frac{ 2 \times 10^{-6} }{ 8 \times 10^{-19} } \amp= \frac{ 2 }{ 8 }\times\frac{ 10^{-6} }{ 10^{-19} } \\ \amp= \highlight{0.25} \times 10^{13} \\ \amp= \highlight{2.5\times 10^{-1}} \times 10^{13} \\ \amp= 2.5 \times 10^{12} \end{align*}

There are times where we will have to raise numbers written in scientific notation to a power. For example, suppose we have to find the area of a square whose radius is \(3\times 10^7\) feet. To perform this calculation, we first remember the formula for the area of a square, \(A=s^2\) and then substitute \(3\times 10^7\) for \(s\text{:}\) \(A = \left( 3\times 10^7 \right)^2\text{.}\) To perform this calculation, we'll need to remember to use the product to a power rule and the power to a power rule:

\begin{align*} A \amp= \left( 3\times 10^7 \right)^2\\ \amp= \left( 3\right)^2 \times \left(10^7 \right)^2\\ \amp= 9 \times 10^{14} \end{align*}

SubsectionExercises

Simplifying Products and Quotients Involving Exponents

1

Use the properties of exponents to simplify the expression.

\({y^{18}}\cdot{y^{16}}\)

2

Use the properties of exponents to simplify the expression.

\({t^{2}}\cdot{t^{9}}\)

3

Use the properties of exponents to simplify the expression.

\(\left(y^{3}\right)^{2}\)

4

Use the properties of exponents to simplify the expression.

\(\left(r^{4}\right)^{9}\)

5

Use the properties of exponents to simplify the expression.

\(\left(2y^{5}\right)^3\)

6

Use the properties of exponents to simplify the expression.

\(\left(4x^{6}\right)^2\)

7

Use the properties of exponents to simplify the expression.

\(({10x^{12}})\cdot({5x^{14}})\)

8

Use the properties of exponents to simplify the expression.

\(({6y^{14}})\cdot({-4y^{7}})\)

9

Use the properties of exponents to simplify the expression.

\(\displaystyle{\left({-\frac{y^{16}}{8}}\right) \cdot \left({\frac{y^{20}}{5}}\right)}\)

10

Use the properties of exponents to simplify the expression.

\(\displaystyle{\left({\frac{r^{18}}{4}}\right) \cdot \left({-\frac{r^{13}}{4}}\right)}\)

11

Use the properties of exponents to simplify the expression.

\(-3\left(-5y^{12}\right)^2\)

12

Use the properties of exponents to simplify the expression.

\(-4\left(-10y^{3}\right)^3\)

13

Use the properties of exponents to simplify the expression.

\(\left(-40\right)^0=\)

14

Use the properties of exponents to simplify the expression.

\(\left(-34\right)^0=\)

15

Use the properties of exponents to simplify the expression.

\(-23^0=\)

16

Use the properties of exponents to simplify the expression.

\(-28^0=\)

17

Use the properties of exponents to simplify the expression.

\(34^0+\left(-34\right)^0=\)

18

Use the properties of exponents to simplify the expression.

\(39^0+\left(-39\right)^0=\)

19

Use the properties of exponents to simplify the expression.

\(45c^0=\)

20

Use the properties of exponents to simplify the expression.

\(50y^0=\)

21

Use the properties of exponents to simplify the expression.

\(\left(-792n\right)^0=\)

22

Use the properties of exponents to simplify the expression.

\(\left(-571c\right)^0=\)

23

Use the properties of exponents to simplify the expression.

\(\left(\displaystyle\frac{x^{2}}{4}\right)^{3}=\)

24

Use the properties of exponents to simplify the expression.

\(\left(\displaystyle\frac{x^{8}}{5}\right)^{2}=\)

25

Use the properties of exponents to simplify the expression.

\(\left(\displaystyle\frac{-7}{4x^{6}}\right)^{3}=\)

26

Use the properties of exponents to simplify the expression.

\(\left(\displaystyle\frac{-7}{2x^{10}}\right)^{3}=\)

27

Use the properties of exponents to simplify the expression.

\(\left(\displaystyle\frac{5x^{8}}{4}\right)^{3}=\)

28

Use the properties of exponents to simplify the expression.

\(\left(\displaystyle\frac{9x^{9}}{10}\right)^{2}=\)

29

Use the properties of exponents to simplify the expression.

\(\displaystyle\frac{{-30t^{15}}}{{15t}}=\)

30

Use the properties of exponents to simplify the expression.

\(\displaystyle\frac{{15y^{10}}}{{3y}}=\)

31

Use the properties of exponents to simplify the expression.

\(\displaystyle\frac{{4x^{3}}}{{12x}}=\)

32

Use the properties of exponents to simplify the expression.

\(\displaystyle\frac{{6x^{10}}}{{36x^{8}}}=\)

33

Use the properties of exponents to simplify the expression.

\(\displaystyle\frac{{t^{10}}}{{t^{6}}}=\)

34

Use the properties of exponents to simplify the expression.

\(\displaystyle\frac{{y^{12}}}{{y^{3}}}=\)

35

Use the properties of exponents to simplify the expression.

\(\displaystyle\frac{{16^{20}}}{{16^{13}}}=\)

36

Use the properties of exponents to simplify the expression.

\(\displaystyle\frac{{18^{16}}}{{18^{3}}}=\)

37

Use the properties of exponents to simplify the expression.

\(\left(\displaystyle\frac{x^{4}}{2y^{6}z^{10}}\right)^{3}=\)

38

Use the properties of exponents to simplify the expression.

\(\left(\displaystyle\frac{x^{9}}{2y^{10}z^{3}}\right)^{3}=\)

39

Use the properties of exponents to simplify the expression.

\(\left(\displaystyle\frac{-9x^{2}}{10y^{6}}\right)^{2}=\)

40

Use the properties of exponents to simplify the expression.

\(\left(\displaystyle\frac{-5x^{3}}{2y^{8}}\right)^{3}=\)

41

Use the properties of exponents to simplify the expression.

\(\displaystyle\frac{{12^{18}}\cdot{13^{19}}}{{12^{4}}\cdot{13^{16}}}=\)

42

Use the properties of exponents to simplify the expression.

\(\displaystyle\frac{{14^{12}}\cdot{10^{19}}}{{14^{7}}\cdot{10^{8}}}=\)

43

Use the properties of exponents to simplify the expression.

\(\displaystyle\frac{{-24x^{18}y^{19}z^{5}}}{{12x^{16}y^{10}z^{3}}}=\)

44

Use the properties of exponents to simplify the expression.

\(\displaystyle\frac{{-70x^{15}y^{11}z^{10}}}{{14x^{13}y^{7}z^{5}}}=\)

45

Use the properties of exponents to simplify the expression.

\(\displaystyle\frac{{64x^{7}y^{18}}}{{16x^{6}y^{11}}}=\)

46

Use the properties of exponents to simplify the expression.

\(\displaystyle\frac{{-36x^{20}y^{9}}}{{18x^{17}y^{6}}}=\)

Simplify and write your answer without using negative exponents. All variables represent non-zero real numbers.

47

Rewrite the expression simplified and using only positive exponents.

\(\displaystyle{ \displaystyle\left(\frac{1}{10}\right)^{-3}= }\) \(\quad\)

48

Rewrite the expression simplified and using only positive exponents.

\(\displaystyle{ \displaystyle\left(\frac{1}{2}\right)^{-2}= }\) \(\quad\)

49

Rewrite the expression simplified and using only positive exponents.

\(\displaystyle\frac{3^{-2}}{2^{-3}}=\)

50

Rewrite the expression simplified and using only positive exponents.

\(\displaystyle\frac{4^{-2}}{6^{-3}}=\)

51

Rewrite the expression simplified and using only positive exponents.

\(5^{-1}-7^{-1}=\)

52

Rewrite the expression simplified and using only positive exponents.

\(6^{-1}-2^{-1}=\)

53

Rewrite the expression simplified and using only positive exponents.

\(\displaystyle{\displaystyle {15x^{-9}}= }\)

54

Rewrite the expression simplified and using only positive exponents.

\(\displaystyle{\displaystyle {10x^{-10}}= }\)

55

Rewrite the expression simplified and using only positive exponents.

\(\displaystyle{\displaystyle {\frac{20}{x^{-11}}}= }\)

56

Rewrite the expression simplified and using only positive exponents.

\(\displaystyle{\displaystyle {\frac{14}{x^{-12}}}= }\)

57

Rewrite the expression simplified and using only positive exponents.

\(\displaystyle{\displaystyle {\frac{9x^{-3}}{x}}= }\)

58

Rewrite the expression simplified and using only positive exponents.

\(\displaystyle{\displaystyle {\frac{19x^{-5}}{x}}= }\)

59

Rewrite the expression simplified and using only positive exponents.

\(\displaystyle{\displaystyle {\frac{15x^{-8}}{x^{-17}}}= }\)

60

Rewrite the expression simplified and using only positive exponents.

\(\displaystyle{\displaystyle {\frac{6x^{-10}}{x^{-8}}}= }\)

61

Rewrite the expression simplified and using only positive exponents.

\(\displaystyle{\displaystyle {\frac{13x^{-17}}{14x^{-33}}}= }\)

62

Rewrite the expression simplified and using only positive exponents.

\(\displaystyle{\displaystyle {\frac{19x^{-21}}{20x^{-23}}}= }\)

63

Rewrite the expression simplified and using only positive exponents.

\(\displaystyle\frac{t^{-8}}{x^{-7}}=\)

64

Rewrite the expression simplified and using only positive exponents.

\(\displaystyle\frac{t^{-13}}{x^{-8}}=\)

65

Rewrite the expression simplified and using only positive exponents.

\(\displaystyle\frac{t^{-5}}{y^{15}}=\)

66

Rewrite the expression simplified and using only positive exponents.

\(\displaystyle\frac{x^{-10}}{r^{12}}=\)

67

Rewrite the expression simplified and using only positive exponents.

\(\displaystyle\frac{1}{41x^{-2}}=\)

68

Rewrite the expression simplified and using only positive exponents.

\(\displaystyle\frac{1}{24y^{-10}}=\)

69

Rewrite the expression simplified and using only positive exponents.

\(\displaystyle\frac{y^{6}}{y^{7}}=\)

70

Rewrite the expression simplified and using only positive exponents.

\(\displaystyle\frac{r^{11}}{r^{39}}=\)

71

Rewrite the expression simplified and using only positive exponents.

\(\displaystyle\frac{12r^{15}}{2r^{22}}=\)

72

Rewrite the expression simplified and using only positive exponents.

\(\displaystyle\frac{-54t^{2}}{9t^{5}}=\)

73

Rewrite the expression simplified and using only positive exponents.

\(\displaystyle\frac{-9t^{19}}{13t^{37}}=\)

74

Rewrite the expression simplified and using only positive exponents.

\(\displaystyle\frac{-5t^{18}}{7t^{20}}=\)

75

Rewrite the expression simplified and using only positive exponents.

\(\displaystyle\frac{x^{2}}{\left(x^{3}\right)^{10}}=\)

76

Rewrite the expression simplified and using only positive exponents.

\(\displaystyle\frac{x^{7}}{\left(x^{9}\right)^{8}}=\)

77

Rewrite the expression simplified and using only positive exponents.

\(\displaystyle\frac{y^{-2}}{\left(y^{6}\right)^{5}}=\)

78

Rewrite the expression simplified and using only positive exponents.

\(\displaystyle\frac{y^{-7}}{\left(y^{12}\right)^{3}}=\)

79

Rewrite the expression simplified and using only positive exponents.

\(r^{-14}\cdot r^{2}=\)

80

Rewrite the expression simplified and using only positive exponents.

\(r^{-8}\cdot r^{4}=\)

81

Rewrite the expression simplified and using only positive exponents.

\((-4r^{-19})\cdot (9r^{16})=\)

82

Rewrite the expression simplified and using only positive exponents.

\((-2t^{-13})\cdot (4t^{4})=\)

83

Rewrite the expression simplified and using only positive exponents.

\(\displaystyle\left(\frac{8}{5}\right)^{-2}=\)

84

Rewrite the expression simplified and using only positive exponents.

\(\displaystyle\left(\frac{9}{2}\right)^{-2}=\)

85

Rewrite the expression simplified and using only positive exponents.

\(\left(-3\right)^{-3}=\)

86

Rewrite the expression simplified and using only positive exponents.

\(\left(-4\right)^{-2}=\)

87

Rewrite the expression simplified and using only positive exponents.

\(\displaystyle\frac{1}{(-5)^{-3}}=\)

88

Rewrite the expression simplified and using only positive exponents.

\(\displaystyle\frac{1}{(-6)^{-2}}=\)

89

Rewrite the expression simplified and using only positive exponents.

\(\displaystyle\frac{-3}{(-3)^{-2}}=\)

90

Rewrite the expression simplified and using only positive exponents.

\(\displaystyle\frac{9}{(-4)^{-3}}=\)

91

Rewrite the expression simplified and using only positive exponents.

\(9^{-2}=\)

92

Rewrite the expression simplified and using only positive exponents.

\(10^{-2}=\)

93

Rewrite the expression simplified and using only positive exponents.

\(2^{-1}+8^{-1}=\)

94

Rewrite the expression simplified and using only positive exponents.

\(3^{-1}+5^{-1}=\)

95

Rewrite the expression simplified and using only positive exponents.

\(\displaystyle\frac{1}{4^{-2}}=\)

96

Rewrite the expression simplified and using only positive exponents.

\(\displaystyle\frac{1}{5^{-3}}=\)

97

Rewrite the expression simplified and using only positive exponents.

\(-6^{-2}=\)

98

Rewrite the expression simplified and using only positive exponents.

\(-7^{-2}=\)

99

Rewrite the expression simplified and using only positive exponents.

\(\displaystyle\frac{\left(2r^{9}\right)^{2}}{r^{25}}=\)

100

Rewrite the expression simplified and using only positive exponents.

\(\displaystyle\frac{\left(2t^{6}\right)^{3}}{t^{19}}=\)

101

Rewrite the expression simplified and using only positive exponents.

\(\displaystyle\frac{\left(2t^{12}\right)^{2}}{t^{-12}}=\)

102

Rewrite the expression simplified and using only positive exponents.

\(\displaystyle\frac{\left(2x^{9}\right)^{3}}{x^{-8}}=\)

103

Rewrite the expression simplified and using only positive exponents.

\(\displaystyle\left(\frac{x^{8}}{x^{2}}\right)^{-2}=\)

104

Rewrite the expression simplified and using only positive exponents.

\(\displaystyle\left(\frac{y^{19}}{y^{9}}\right)^{-5}=\)

105

Rewrite the expression simplified and using only positive exponents.

\(\displaystyle\left(\frac{10y^{13}}{2y^{11}}\right)^{-4}=\)

106

Rewrite the expression simplified and using only positive exponents.

\(\displaystyle\left(\frac{6r^{7}}{2r^{3}}\right)^{-3}=\)

107

Rewrite the expression simplified and using only positive exponents.

\(\left(-5r^{-19}\right)^{-3}\)

108

Rewrite the expression simplified and using only positive exponents.

\(\left(-2r^{-12}\right)^{-2}\)

109

Rewrite the expression simplified and using only positive exponents.

\(\left(4t^{-6}\right)^{-3}\)

110

Rewrite the expression simplified and using only positive exponents.

\(\left(3t^{-18}\right)^{-3}\)

111

Rewrite the expression simplified and using only positive exponents.

\(\displaystyle\frac{9x^{6}\cdot10x^{5}}{7x^{10}}=\)

112

Rewrite the expression simplified and using only positive exponents.

\(\displaystyle\frac{10x^{3}\cdot3x^{8}}{7x^{8}}=\)

113

Rewrite the expression simplified and using only positive exponents.

\(\left(y^{13}\right)^{2}\cdot y^{-12}=\)

114

Rewrite the expression simplified and using only positive exponents.

\(\left(y^{8}\right)^{4}\cdot y^{-6}=\)

115

Rewrite the expression simplified and using only positive exponents.

\(\left(4r^{4}\right)^{2}\cdot r^{-7}=\)

116

Rewrite the expression simplified and using only positive exponents.

\(\left(4r^{12}\right)^{3}\cdot r^{-22}=\)

117

Rewrite the expression simplified and using only positive exponents.

\(\displaystyle\frac{\left(r^{6}\right)^{4}}{\left(r^{7}\right)^{5}}=\)

118

Rewrite the expression simplified and using only positive exponents.

\(\displaystyle\frac{\left(t^{3}\right)^{2}}{\left(t^{3}\right)^{5}}=\)

119

Rewrite the expression simplified and using only positive exponents.

\(\left(t^{15}\right)^{-4}=\)

120

Rewrite the expression simplified and using only positive exponents.

\(\left(x^{9}\right)^{-2}=\)

121

Rewrite the expression simplified and using only positive exponents.

\(\left(x^{5}r^{14}\right)^{-4}=\)

122

Rewrite the expression simplified and using only positive exponents.

\(\left(y^{15}r^{15}\right)^{-5}=\)

123

Rewrite the expression simplified and using only positive exponents.

\(\left(y^{-13}r^{11}\right)^{-4}=\)

124

Rewrite the expression simplified and using only positive exponents.

\(\left(r^{-13}t^{7}\right)^{-5}=\)

125

Rewrite the expression simplified and using only positive exponents.

\(\displaystyle\left(\frac{r^{10}}{2}\right)^{-2}=\)

126

Rewrite the expression simplified and using only positive exponents.

\(\displaystyle\left(\frac{r^{6}}{4}\right)^{-3}=\)

127

Rewrite the expression simplified and using only positive exponents.

\(\displaystyle\left(\frac{t^{10}}{x^{13}}\right)^{-4}=\)

128

Rewrite the expression simplified and using only positive exponents.

\(\displaystyle\left(\frac{t^{8}}{r^{6}}\right)^{-5}=\)

129

Rewrite the expression simplified and using only positive exponents.

\(\displaystyle\frac{\left(x^{8}t^{-7}\right)^{-3}}{\left(x^{-3}t^{8}\right)^{-2}}=\)

130

Rewrite the expression simplified and using only positive exponents.

\(\displaystyle\frac{\left(x^{4}t^{-7}\right)^{-4}}{\left(x^{-3}t^{4}\right)^{-2}}=\)

131

Rewrite the expression simplified and using only positive exponents.

\(8x^{-5}y^{5}z^{-4}\left(3x^{8}\right)^{-2}=\)

132

Rewrite the expression simplified and using only positive exponents.

\(2x^{-3}y^{4}z^{-2}\left(3x^{2}\right)^{-2}=\)

133

Rewrite the expression simplified and using only positive exponents.

\(\displaystyle \left( \frac{x^{5}y^{4}z^{8}}{x^{-6}y^{-5}z^{-2}}\right)^{-2}=\)

134

Rewrite the expression simplified and using only positive exponents.

\(\displaystyle \left( \frac{x^{6}y^{8}z^{6}}{x^{-5}y^{-4}z^{-5}}\right)^{-2}=\)

Convert the numbers into scientific notation.

135

Write the following number in scientific notation.

\(750=\)

136

Write the following number in scientific notation.

\(850000=\)

137

Write the following number in scientific notation.

\(9500=\)

138

Write the following number in scientific notation.

\(150=\)

139

Write the following number in scientific notation.

\(0.026=\)

140

Write the following number in scientific notation.

\(0.0035=\)

141

Write the following number in scientific notation.

\(0.00045=\)

142

Write the following number in scientific notation.

\(0.055=\)

Convert the numbers into decimals not in scientific notation.

143

Write the following number in decimal notation without using exponents.

\(6.5\times 10^{3}=\)

144

Write the following number in decimal notation without using exponents.

\(7.5\times 10^{2}=\)

145

Write the following number in decimal notation without using exponents.

\(8.51\times 10^{4}=\)

146

Write the following number in decimal notation without using exponents.

\(9.51\times 10^{3}=\)

147

Write the following number in decimal notation without using exponents.

\(1.49\times 10^{0}=\)

148

Write the following number in decimal notation without using exponents.

\(2.49\times 10^{0}=\)

149

Write the following number in decimal notation without using exponents.

\(3.5\times 10^{-4}=\)

150

Write the following number in decimal notation without using exponents.

\(4.5\times 10^{-2}=\)

151

Write the following number in decimal notation without using exponents.

\(5.48\times 10^{-3}=\)

152

Write the following number in decimal notation without using exponents.

\(6.48\times 10^{-4}=\)

Perform the indicated operation. Write your answers using scientific notation.

153

Multiply the following numbers, writing your answer in scientific notation.

\((7\times 10^{5})(8\times 10^{3})=\)

154

Multiply the following numbers, writing your answer in scientific notation.

\((8\times 10^{4})(4\times 10^{2})=\)

155

Divide the following numbers, writing your answer in scientific notation.

\(\displaystyle\frac{2.7\times 10^{5}}{9\times 10^{3}}=\)

156

Divide the following numbers, writing your answer in scientific notation.

\(\displaystyle\frac{1.6\times 10^{3}}{2\times 10^{4}}=\)

157

Divide the following numbers, writing your answer in scientific notation.

\(\displaystyle\frac{1.8\times 10^{5}}{3\times 10^{2}}=\)

158

Divide the following numbers, writing your answer in scientific notation.

\(\displaystyle\frac{1.2\times 10^{6}}{4\times 10^{5}}=\)

159

Divide the following numbers, writing your answer in scientific notation.

\(\displaystyle\frac{4\times 10^{3}}{5\times 10^{-4}}=\)

160

Divide the following numbers, writing your answer in scientific notation.

\(\displaystyle\frac{2.5\times 10^{5}}{5\times 10^{-4}}=\)

161

Divide the following numbers, writing your answer in scientific notation.

\(\displaystyle\frac{1.2\times 10^{2}}{6\times 10^{-5}}=\)

162

Divide the following numbers, writing your answer in scientific notation.

\(\displaystyle\frac{5.6\times 10^{4}}{7\times 10^{-2}}=\)

163

Divide the following numbers, writing your answer in scientific notation.

\(\displaystyle\frac{4\times 10^{-5}}{8\times 10^{4}}=\)

164

Divide the following numbers, writing your answer in scientific notation.

\(\displaystyle\frac{1.8\times 10^{-4}}{9\times 10^{3}}=\)

165

Divide the following numbers, writing your answer in scientific notation.

\(\displaystyle\frac{1.4\times 10^{-2}}{2\times 10^{2}}=\)

166

Divide the following numbers, writing your answer in scientific notation.

\(\displaystyle\frac{1.5\times 10^{-4}}{3\times 10^{4}}=\)

167

Simplify the following expression, writing your answer in scientific notation.

\((3\times 10^{2})^{4}=\)

168

Simplify the following expression, writing your answer in scientific notation.

\((3\times 10^{8})^{2}=\)