3. Review Exercises

Answers to the odd-numbered questions are available at the end of the book.

1. Calculate the difference between [latex]2^5[/latex] and [latex]5^2[/latex].
2. Calculate the difference between [latex]3^4[/latex] and [latex]4^3[/latex].
3. Express 243 as a power of 3 and then evaluate [latex]\displaystyle{243\frac{3}{5}}[/latex].
4. Express 512 as a power of 2 and then evaluate [latex]\displaystyle{512\frac{4}{9}}[/latex].

Express Problems 5 to 8 as a single power and then evaluate.

5. a. [latex]\displaystyle{(2^6)^{\frac{1}{3}}}[/latex]
   b. [latex]\displaystyle{(5^{15})^{\frac{1}{5}}}[/latex]
6. a. [latex]\displaystyle{\left(\frac{3^9}{3^3}\right)^{\frac{1}{3}}}[/latex]
   b. [latex]\displaystyle{\left(\frac{2^{12}}{2^4}\right)^{\frac{1}{4}}}[/latex]
7. a. [latex]\displaystyle{(3^2)^{\frac{1}{2}} \times (3^3)^{\frac{2}{3}}}[/latex]
   b. [latex]\displaystyle{(6^2)^{\frac{1}{3}} \times (6^3)^{\frac{1}{9}}}[/latex]
8. a. [latex]\displaystyle{(2^2)^{\frac{1}{4}} \times (2^5)^{\frac{3}{10}}}[/latex]
   b. [latex]\displaystyle{(5^3)^{\frac{2}{3}} \times (5^2)^{\frac{1}{2}}}[/latex]

For Problems 9 to 16, simplify using laws of exponents and then evaluate.

9. a. [latex]\displaystyle{\frac{2^3 \times 3^4 \times 2^2}{3 \times 2^5}}[/latex]
   b. [latex]\displaystyle{\frac{(5^2) \times 5^4}{5^7}}[/latex]
10. a. [latex]\displaystyle{\frac{5^2 \times 7^3 \times 54}{7 \times 5^6}}[/latex]
    b. [latex]\displaystyle{\frac{(2^5) \times 2^2}{2^{17}}}[/latex]
11. a. [latex](-5)^2 \times (4)^2[/latex]
    b. [latex]-10^4 \times 10^3[/latex]
12. a. [latex](-2)^2 \times (3)^2[/latex]
    b. [latex]-2^4 \times 2^2[/latex]
13. a. [latex]\displaystyle{(125)^{-\frac{1}{3}}}[/latex]
    b. [latex]\displaystyle{(49)^{-\frac{1}{2}}}[/latex]
    c. [latex]\displaystyle{\sqrt{\frac{64}{81}}}[/latex]
14. a. [latex]\displaystyle{(16)^{-\frac{1}{4}}}[/latex]
    b. [latex]\displaystyle{(27)^{-\frac{1}{3}}}[/latex]
    c. [latex]\displaystyle{\sqrt{\frac{25}{49}}}[/latex]
15. a. [latex]\displaystyle{\sqrt{7^4}}[/latex]
    b. [latex]\displaystyle{\sqrt{\frac{25}{36}}}[/latex]
    c. [latex]\displaystyle{\sqrt[3]{\frac{216}{125}}}[/latex]
16. a. [latex]\displaystyle{\sqrt{5^6}}[/latex]
    b. [latex]\displaystyle{\sqrt{\frac{49}{64}}}[/latex]
    c. [latex]\displaystyle{\sqrt[3]{\frac{64}{27}}}[/latex]

Evaluate Problems 17 to 22 and express the answers rounded to two decimal places, wherever applicable.

17. a. [latex]\displaystyle{6,000\left(1 + \frac{0.06}{12}\right)^{36}}[/latex]
    b. [latex]2,000(1 + 0.004)^{-24}[/latex]
18. a. [latex]\displaystyle{4,000\left(1 + \frac{0.075}{12}\right)^{60}}[/latex]
    b. [latex]5,000(1 + 0.003)^{-48}[/latex]
19. [latex]\displaystyle{\frac{3,000[(1.06)^{25} - 1]}{0.06}}[/latex]
20. [latex]\displaystyle{\frac{1,400[(1.03)^{30} - 1]}{0.03}}[/latex]
21. [latex]\displaystyle{\frac{950[1 - (1.03)^{15}]}{0.03}}[/latex]
22. [latex]\displaystyle{\frac{1,200[1 - (1.04)^{20}]}{0.04}}[/latex]