Exponents and Radicals

Here you will find a concise collection of exponents and radicals practice problems. Visit this page directly at hunkim.com/exponents

1. $3^3$
   Solution
   27
2. $2^{10}$
   Solution
   1024
3. $(-3)^3$
   Solution
   $-27$
4. $(-2)^4$
   Solution
   16
5. $1^{123}$
   Solution
   1
6. $1^0$
   Solution
   1
7. $1^{2.5+\pi \sqrt{5}}$
   Solution
   1
8. $0^1$
   Solution
   $0$
9. $0^0$
   Solution
   Undefined
10. $(-1)^{101}$
    Solution
    $-1$
11. $-(-1)^{10}$
    Solution
    $-1$
12. $\left (\frac{5}{3}\right)^2$
    Solution
    25/9
13. $\left( \frac{5}{3}\right)^{-2}$
    Solution
    9/25
14. $(p^3)(p^2)\times p$
    Solution
    $p^6$
15. $(x^2)^3$
    Solution
    $x^6$
16. $(2a^2b^3)^3$
    Solution
    $8a^6 b^9$
17. $\frac{9x^4y^4}{3xy^2}$
    Solution
    $3x^3 y^2$
18. $\frac{x}{x^{-4}}$
    Solution
    $x^5$
19. $\left( \frac{4pq^6}{2p^3q^3}\right)^3$
20. Solution
    $\frac{8q^9}{p^6}$
21. $\left( \frac{a}{b^3}\right)^{-2}$
    Solution
    $\frac{b^6}{a^2}$
22. Solve $\frac{3^{10}}{3^x}=3^6$
    Solution
    4
23. Solve $\left( \frac{2^{1-2x}}{2^{x+3}}\right)^3=4$
    Solution
    $-8/9$
24. Write $\frac{15^6}{3^6}$ as a single power
    Solution
    15625
25. Solve $2^{x+1}=8^{3-2x}$
    Solution
    8/7
26. Simplify $-\frac{8x^5-12x^3}{6x^3}$
    Solution
    $2-\frac{4}{3} x^2$
27. Express $x^{1/2}$ as a radical
28. Solution
    $\sqrt{x}$
29. Express $\sqrt[3]{x}$ as a fractional exponent
    Solution
    $x^{1/3}$
30. Write $x^{2/5}$ as a radical
    Solution
    $\sqrt[5]{x^2}$
31. Evaluate $\sqrt{\frac{9}{25}}$
    Solution
    $\frac{3}{5}$
32. Write $\sqrt{8}$ as a mixed radical
    Solution
    $2\sqrt{2}$
33. Write $\sqrt[3]{81}$ as a mixed radical
    Solution
    $3\sqrt[3]{3}$
34. Write $3\sqrt{2}$ as an entire radical
    Solution
    $\sqrt{18}$
35. Write $2\sqrt[3]{3}$ as an entire radical
    Solution
    $\sqrt[3]{24}$
36. Simplify $\sqrt{2}\times \sqrt{5}+\left( \sqrt[3]{2}\right) \left( \sqrt[3]{5}\right)$
    Solution
    $\sqrt{10}+\sqrt[3]{10}$
37. Simplify $\frac{\sqrt{125}}{5}$
    Solution
    $\sqrt{5}$
38. $(0.09)^{1/2}$
    Solution
    0.3
39. $(0.000008)^{1/3}$
    Solution
    0.02
40. Solve $\sqrt{ \sqrt[3]{ \sqrt{x}} }=2$
    Solution
    4096
41. Rationalize:
    1. $\frac{1}{\sqrt{2}}$
       Solution
       $\sqrt{2}/2$
    2. $\frac{3}{\sqrt{3}}$
       Solution
       $\sqrt{3}/3$
    3. $\frac{2}{\sqrt[3]{2}}$
       Solution
       $\sqrt[3]{4}$
42. Challenge:
    1. Challenge:$\frac{(0.6x^{-1})^{-2}}{\left( \frac{2}{x}\right)^3}$
       Solution
       $\frac{25x^5}{72}$
    2. Solve $9\times 27+2(3)^x=3^{x+1}$
       Solution
       5
    3. Show that $\sqrt[c]{a^b}=\left( \sqrt[c]{a}\right)^b$
       Solution
       $LS=\left( a^b \right)^{\frac{1}{c}}=a^{\frac{b}{c}}$
       $RS=\left( a^{\frac{1}{c}} \right)^b=a^{\frac{b}{c}}=LS$
       
    4. Solve $9\times 27+2(3)^x=3^{x+1}$
       Solution
       5
    5. Solve $4^{1.5x}=2(32^{x-1})$
       Solution
       2
    6. Solve $x^{2/3}=5$
       Solution
       $5^{\frac{3}{2}}=5\sqrt{5}$

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