ORCCA More Exponent Rules

Section 5.6 More Exponent Rules

Objectives: PCC Course Content and Outcome Guide

MTH 65 CCOG 1.a

Figure 5.6.1. Alternative Video Lesson

Subsection 5.6.1 Review of Exponent Rules for Products and Exponents

In Section 5.2, we introduced three basic rules involving products and exponents. Then in Section 5.3, we introduced one more. We begin this section with a recap of these four exponent rules.

List 5.6.2. Summary of Exponent Rules (Thus Far)

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*}

Quotient of Powers Rule

When dividing two expressions that have the same base, simplify the quotient by subtracting the exponents.

\begin{equation*} \frac{x^m}{x^n} = x^{m-n} \end{equation*}

For now, we only know this rule when \(m\gt n\text{.}\)

Checkpoint 5.6.3.

Subsection 5.6.2 Quotient to a Power Rule

One rule we have learned is the product to a power rule, as in \((2x)^{3}=2^{3}x^{3}\text{.}\) When two factors are multiplied and the product is raised to a power, we may apply the exponent to each of those factors individually. We can use the rules of fractions to extend this property to a quotient raised to a power.

Example 5.6.4.

Let \(y\) be a real number, where \(y \neq 0\text{.}\) Find another way to write \(\left(\frac{5}{y}\right)^4\text{.}\)

Explanation

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

\begin{align*} \left( \frac{5}{y} \right)^4 \amp= \left( \frac{5}{y} \right) \left( \frac{5}{y} \right) \left( \frac{5}{y} \right) \left( \frac{5}{y} \right)\\ \amp= \frac{5 \cdot 5 \cdot 5 \cdot 5}{y \cdot y \cdot y \cdot y}\\ \amp= \frac{5^4}{y^4}\\ \amp= \frac{625}{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:

Fact 5.6.5. Quotient to a Power Rule.

For real numbers \(a\) and \(b\) (with \(b \neq 0\)) and natural number \(m\text{,}\)

\begin{equation*} \left( \frac{a}{b} \right)^{m} = \frac{a^{m}}{b^{m}} \end{equation*}

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 5.6.4, this means that we can directly calculate \(\left( \frac{5}{y} \right)^4\text{:}\)

\begin{align*} \left( \frac{5}{y} \right)^4 \amp= \frac{5^4}{y^4}\\ \amp=\frac{625}{y^4} \end{align*}

Checkpoint 5.6.6.

Subsection 5.6.3 Zero as an Exponent

So far, we have been working with exponents that are natural numbers (\(1, 2, 3, \ldots\)). By the end of this section, we will expand our understanding to include exponents that are any integer, as with \(5^{0}\) and \(12^{-2}\text{.}\) As a first step, let's explore how \(0\) should behave as an exponent by considering the pattern of decreasing powers of \(2\) in Figure 5.6.7.

Power		Product		Result
\(2^4\)	\(=\)	\(2 \cdot 2 \cdot 2 \cdot 2\)	\(=\)	\(16\)
\(2^3\)	\(=\)	\(2 \cdot 2 \cdot 2\)	\(=\)	\(8\)	(divide by \(2\))
\(2^2\)	\(=\)	\(2 \cdot 2\)	\(=\)	\(4\)	(divide by \(2\))
\(2^1\)	\(=\)	\(2\)	\(=\)	\(2\)	(divide by \(2\))
\(2^0\)	\(=\)	\(\mathord{?}\)	\(=\)	\(\mathord{?}\)

Figure 5.6.7. Descending Powers of \(2\)

As we move down from one row to the row below it, we reduce the exponent in the power by \(1\) and we remove a factor of \(2\) from the product. The result in one row is half of the result of the previous row. The question is, what happens when the exponent gets down to \(0\) and you remove the last remaining factor of \(2\text{?}\) Following that pattern with the final results, moving from \(2^1\) to \(2^0\) should meant the result of \(2\) is divided by \(2\text{,}\) leaving \(1\text{.}\) So we have:

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

Fact 5.6.8. The Zero Exponent Rule.

For any non-zero real number \(a\text{,}\)

\begin{equation*} a^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 \(0\text{.}\)

Checkpoint 5.6.9.

Subsection 5.6.4 Negative Exponents

We understand what it means for a variable to have a natural number exponent. For example, \(x^5\) means \(\overbrace{x\cdot x\cdot x\cdot x\cdot x}^{\text{five times}}\text{.}\) Now we will try to give meaning to an exponent that is a negative integer, like in \(x^{-5}\text{.}\)

To consider what it could possibly mean to have a negative integer exponent, let's extend the pattern we examined in Figure 5.6.7. 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, going all the way down into when the exponent is negeative.

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

Figure 5.6.10. Negative Powers of \(2\)

We are seeing a pattern where \(2^{\text{negative number}}\) is equal to \(\frac{1}{2^{\text{positive number}}}\text{.}\) 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.

Fact 5.6.11. The Negative Exponent Rule.

For any non-zero real number \(a\) and any natural number \(n\text{,}\)

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

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 power in the numerator with a negative exponent belongs in the denominator (with a positive exponent). And similarly, a power in the denominator with a negative exponent belongs in the numerator (with a positive exponent). In other words, you can view a negative exponent as telling you to move something to/from the numerator/denominator of an expression, changing the sign of the exponent at the same time.

You may be expected to simplify expressions so that they do not have any negative exponents. This can always be accomplished using the negative exponent rule. Try it with these exercises.

Checkpoint 5.6.12.

Subsection 5.6.5 Summary of Exponent Rules

Now that we have some new exponent rules beyond those from Section 5.2 and Section 5.3, let's summarize.

List 5.6.13. Summary of the Rules of Exponents for Multiplication and Division

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

Product Rule: \(a^{m} \cdot a^{n} = a^{m+n}\)
Power to a Power Rule: \((a^{m})^{n} = a^{m\cdot n}\)
Product to a Power Rule: \((ab)^{m} = a^{m} \cdot b^{m}\)
Quotient Rule: \(\dfrac{a^{m}}{a^{n}} = a^{m-n}\text{,}\) as long as \(a \neq 0\)
Quotient to a Power Rule: \(\left( \dfrac{a}{b} \right)^{m} = \dfrac{a^{m}}{b^{m}}\text{,}\) as long as \(b \neq 0\)
Zero Exponent Rule: \(a^{0} = 1\) for \(a\neq0\)
Negative Exponent Rule: \(a^{-m} = \frac{1}{a^m}\)
Negative Exponent Reciprocal Rule: \(\frac{1}{a^{-m}} = a^m\)

Remark 5.6.14. Why we have “\(a \neq 0\)” and “\(b \neq 0\)” for some rules.

We have to be careful to make sure the rules we state don't suggest that it would ever be OK to divide 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 \(a=0\text{.}\)

Warning 5.6.15. A Common Mistake.

It may be tempting to apply the rules of exponents to expressions containing addition or subtraction. However, none of the Summary of the Rules of Exponents for Multiplication and Division 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 \(a^m + a^n = a^{m+n}\text{?}\) How would that work out when \(a=2\text{,}\) \(m=3\text{,}\) and \(n=4\text{?}\)

\begin{align*} 2^3 + 2^4 \amp\stackrel{?}{=} 2^{3+4}\\ 8 + 16 \amp\stackrel{?}{=} 2^{7}\\ 24 \amp\stackrel{\text{no}}{=} 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 \(a^n + a^m\text{.}\)

Checkpoint 5.6.16.

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.

Checkpoint 5.6.17.

Reading Questions 5.6.6 Reading Questions

1.

When you are considering using the exponent rule \(a^m\cdot a^n=a^{m+n}\text{,}\) are \(m\) and \(n\) allowed to be negative integers?

2.

What are the differences between these three expressions?

\begin{equation*} x+0\qquad0x\qquad x^0 \end{equation*}

3.

If you rearrange \(\frac{xy^{-3}}{a^2b^8c}\) so that it is written without negative exponents, how many factors will you have “moved?”

Exercises 5.6.7 Exercises

Review and Warmup

1.

Evaluate the following.

\(3^{2}\)
\(2^{3}\)
\((-4)^{2}\)
\((-2)^{3}\)

2.

Evaluate the following.

\(3^{2}\)
\(5^{3}\)
\((-4)^{2}\)
\((-5)^{3}\)

3.

Use the properties of exponents to simplify the expression.

\({6}\cdot{6^{7}}\)

4.

Use the properties of exponents to simplify the expression.

\({7}\cdot{7^{4}}\)

5.

Use the properties of exponents to simplify the expression.

\({7^{10}}\cdot{7^{8}}\)

6.

Use the properties of exponents to simplify the expression.

\({8^{7}}\cdot{8^{2}}\)

Simplifying Products and Quotients Involving Exponents

7.

Use the properties of exponents to simplify the expression.

\({r^{20}}\cdot{r^{6}}\)

8.

Use the properties of exponents to simplify the expression.

\({t^{3}}\cdot{t^{18}}\)

9.

Use the properties of exponents to simplify the expression.

\(\displaystyle{\left(y^{4}\right)^{7}}\)

10.

Use the properties of exponents to simplify the expression.

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

11.

Use the properties of exponents to simplify the expression.

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

12.

Use the properties of exponents to simplify the expression.

\(\left(4y^{7}\right)^3\)

13.

Use the properties of exponents to simplify the expression.

\(\displaystyle{({-2y^{13}})\cdot({9y^{4}})}\)

14.

Use the properties of exponents to simplify the expression.

\(\displaystyle{({6r^{15}})\cdot({-8r^{16}})}\)

15.

Use the properties of exponents to simplify the expression.

\(\displaystyle{\left({-\frac{r^{18}}{8}}\right) \cdot \left({\frac{r^{10}}{8}}\right)}\)

16.

Use the properties of exponents to simplify the expression.

\(\displaystyle{\left({-\frac{r^{20}}{4}}\right) \cdot \left({-\frac{r^{3}}{7}}\right)}\)

17.

Use the properties of exponents to simplify the expression.

\(-2\left(-4r^{2}\right)^3\)

18.

Use the properties of exponents to simplify the expression.

\(-3\left(-10r^{3}\right)^2\)

19.

Use the properties of exponents to simplify the expression.

\(\left(-36\right)^0=\)

20.

Use the properties of exponents to simplify the expression.

\(\left(-31\right)^0=\)

21.

Use the properties of exponents to simplify the expression.

\(-27^0=\)

22.

Use the properties of exponents to simplify the expression.

\(-32^0=\)

23.

Use the properties of exponents to simplify the expression.

\(37^0+\left(-37\right)^0=\)

24.

Use the properties of exponents to simplify the expression.

\(43^0+\left(-43\right)^0=\)

25.

Use the properties of exponents to simplify the expression.

\(48q^0=\)

26.

Use the properties of exponents to simplify the expression.

\(5B^0=\)

27.

Use the properties of exponents to simplify the expression.

\(\left(-649t\right)^0=\)

28.

Use the properties of exponents to simplify the expression.

\(\left(-428p\right)^0=\)

29.

Use the properties of exponents to simplify the expression.

\(\left(\displaystyle\frac{x^{7}}{5}\right)^{3}=\)

30.

Use the properties of exponents to simplify the expression.

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

31.

Use the properties of exponents to simplify the expression.

\(\left(\displaystyle\frac{-7}{10x^{6}}\right)^{2}=\)

32.

Use the properties of exponents to simplify the expression.

\(\left(\displaystyle\frac{-7}{6x^{10}}\right)^{3}=\)

33.

Use the properties of exponents to simplify the expression.

\(\left(\displaystyle\frac{5x^{9}}{6}\right)^{2}=\)

34.

Use the properties of exponents to simplify the expression.

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

35.

Use the properties of exponents to simplify the expression.

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

36.

Use the properties of exponents to simplify the expression.

\(\left(\displaystyle\frac{x^{3}}{2y^{7}z^{6}}\right)^{2}=\)

37.

Use the properties of exponents to simplify the expression.

\(\left(\displaystyle\frac{-5x^{4}}{8y^{9}}\right)^{2}=\)