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Section 2.6 Additional Practice with Exponents and Scientific Notation

Exercises Exercises

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

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