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Section2.6Additional Practice with Exponents and Scientific Notation

Subsection2.6.1Exercises

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

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