## Section3.6Additional Practice with Exponents and Scientific Notation

### ExercisesExercises

Completely simplify each expression. Your final expression should contain no negative exponents. Assume that all variables represent positive numbers.

###### 1.

$x^8 \cdot x^{21}$

Solution

$x^8 \cdot x^{21}=x^{29}$

###### 2.

$(x^7)^{10}$

Solution

$(x^7)^{10}=x^{70}$

###### 3.

$\frac{x^{17}}{x^{11}}$

Solution

$\frac{x^{17}}{x^{11}}=x^6$

###### 4.

$\frac{x^{14}}{x^{23}}$

Solution

$\frac{x^{14}}{x^{23}}=\frac{1}{x^9}$

###### 5.

$2x^{-31}$

Solution

$2x^{-31}=\frac{2}{x^{31}}$

###### 6.

$\frac{1}{x^{-96}}$

Solution

$\frac{1}{x^{-96}}=x^{96}$

###### 7.

$-x^{-2}$

Solution

$-x^{-2}=-\frac{1}{x^2}$

###### 8.

$(x^4y^{-3})^5$

Solution

$(x^4y^{-3})^5=\frac{x^{20}}{y^{15}}$

###### 9.

$\frac{x^4x^{-6}}{5x^{-2}}$

Solution

$\frac{x^4x^{-6}}{5x^{-2}}=\frac{1}{5}$

###### 10.

$\left(\frac{2x^{-7}}{y^{-2}}\right)^{-6}$

Solution

$\left(\frac{2x^{-7}}{y^{-2}}\right)^{-6}=\frac{x^{42}}{64y^{12}}$

###### 11.

$\left(\frac{4x^{-1}y^{12}z^{-3}}{(x^2y^{-3})^{-6}}\right)^0$

Solution

$\left(\frac{4x^-1y^{12}z^{-3}}{(x^2y^{-3})^{-6}}\right)^0=1$

###### 12.

$\frac{(xy^{-4})^{-2}}{(3x^{-1})^2}$

Solution

$\frac{(xy^{-4})^{-2}}{(3x^{-1})^2}=\frac{y^8}{9}$

Write each number in scientific notation.

###### 13.

$321,000,000$

Solution

$321,000,000=3.21\times 10^{8}$

###### 14.

$-0.00067$

Solution

$-0.00067=-6.7\times 10^{-4}$

###### 15.

$-11,248,760,000$

Solution

$-11,248,760,000=-1.124876\times 10^{10}$

###### 16.

$0.0000000600$

Solution

$0.0000000600=6.00\times 10^{-8}$

Write each number in standard notation.

###### 17.

$3.2\times 10^{-7}$

Solution

$3.2\times 10^{-7}=0.00000032$

###### 18.

$2.7\times 10^{5}$

Solution

$2.7\times 10^{5}=270,000$

###### 19.

$-1.190\times 10^{8}$

Solution

$-1.190\times 10^{8}=-119,000,000$

###### 20.

$-9.99\times 10^{-9}$

Solution

$-9.99\times 10^{-9}=-0.00000000999$

Determine each product or quotient—write the results using both scientific notation and standard notation.

###### 21.

$(4.0\times 10^{-7})(-1.5\times 10^{9})$

Solution

\begin{aligned}[t] (4.0\times 10^{-7})(-1.5\times 10^{9})\amp=-6.0\times 10^2\\ \amp=-600 \end{aligned}

###### 22.

$(1.2\times 10^{-3})(9.0\times 10^{-4})$

Solution

\begin{aligned}[t] (1.20\times 10^{-3})(9.00\times 10^{-4})\amp=1.08\times 10^{-6}\\ \amp=0.00000108 \end{aligned}

###### 23.

$(-2.5\times 10^{13})(-5.0\times 10^{-7})$

Solution

\begin{aligned}[t] (-2.50\times 10^{13})(-5.00\times 10^{-7})\amp=1.25\times 10^{7}\\ \amp=12,500,000 \end{aligned}

###### 24.

$\frac{4.2\times 10^{11}}{6.0\times 10^{7}}$

Solution

\begin{aligned}[t] \frac{4.20\times 10^{11}}{6.00\times 10^{7}}\amp=7.00\times 10^{3}\\ \amp=7,000 \end{aligned}

###### 25.

$\frac{-3.3\times 10^{-6}}{-3.0\times 10^{-3}}$

Solution

\begin{aligned}[t] \frac{-3.3\times 10^{-6}}{-3.0\times 10^{-3}}\amp=1.1\times 10^{-3}\\ \amp=0.0099 \end{aligned}

###### 26.

$\frac{2.00\times 10^{2}}{6.00\times 10^{-1}}$

Solution

\begin{aligned}[t] \frac{2.00\times 10^{2}}{6.00\times 10^{-1}}\amp=3.33\times 10^{2}\\ \amp=333 \end{aligned}