Evaluate the Exponents: A math worksheet for practicing exponent rules with fractions.
Math worksheet with 20 problems to evaluate exponents involving fractions, including negative and positive bases with various powers.
PNG
612×792
6.7 KB
Free · Personal Use
Quality Assured by Worksheets Library Team
Reviewed for educational accuracy and age-appropriateness
ID: #896309
⭐
Show Answer Key & Explanations
Step-by-step solution for: Exponents and Radicals Worksheets | Exponents & Radicals ...
▼
Show Answer Key & Explanations
Step-by-step solution for: Exponents and Radicals Worksheets | Exponents & Radicals ...
To solve the given problems, we need to evaluate each expression involving exponents. Let's go through each problem step by step.
1. Simplify the fraction inside the parentheses:
\[
-\frac{3}{12} = -\frac{1}{4}
\]
2. Square the simplified fraction:
\[
\left(-\frac{1}{4}\right)^2 = \left(-\frac{1}{4}\right) \cdot \left(-\frac{1}{4}\right) = \frac{1}{16}
\]
Answer: \(\frac{1}{16}\)
---
1. Square the fraction:
\[
\left(-\frac{3}{4}\right)^2 = \left(-\frac{3}{4}\right) \cdot \left(-\frac{3}{4}\right) = \frac{9}{16}
\]
Answer: \(\frac{9}{16}\)
---
1. Cube the fraction:
\[
\left(\frac{1}{3}\right)^3 = \left(\frac{1}{3}\right) \cdot \left(\frac{1}{3}\right) \cdot \left(\frac{1}{3}\right) = \frac{1}{27}
\]
Answer: \(\frac{1}{27}\)
---
1. Square the fraction:
\[
\left(-\frac{4}{5}\right)^2 = \left(-\frac{4}{5}\right) \cdot \left(-\frac{4}{5}\right) = \frac{16}{25}
\]
Answer: \(\frac{16}{25}\)
---
1. Rewrite the negative exponent as a reciprocal:
\[
\left(-\frac{2}{3}\right)^{-5} = \left(\frac{3}{-2}\right)^5 = \left(-\frac{3}{2}\right)^5
\]
2. Raise \(-\frac{3}{2}\) to the power of 5:
\[
\left(-\frac{3}{2}\right)^5 = \left(-\frac{3}{2}\right) \cdot \left(-\frac{3}{2}\right) \cdot \left(-\frac{3}{2}\right) \cdot \left(-\frac{3}{2}\right) \cdot \left(-\frac{3}{2}\right) = -\frac{243}{32}
\]
Answer: \(-\frac{243}{32}\)
---
1. Simplify the fraction inside the parentheses:
\[
-\frac{5}{10} = -\frac{1}{2}
\]
2. Square the simplified fraction:
\[
\left(-\frac{1}{2}\right)^2 = \left(-\frac{1}{2}\right) \cdot \left(-\frac{1}{2}\right) = \frac{1}{4}
\]
Answer: \(\frac{1}{4}\)
---
1. Cube the fraction:
\[
\left(-\frac{2}{7}\right)^3 = \left(-\frac{2}{7}\right) \cdot \left(-\frac{2}{7}\right) \cdot \left(-\frac{2}{7}\right) = -\frac{8}{343}
\]
Answer: \(-\frac{8}{343}\)
---
1. Raise \(\frac{1}{2}\) to the power of 6:
\[
\left(\frac{1}{2}\right)^6 = \left(\frac{1}{2}\right) \cdot \left(\frac{1}{2}\right) \cdot \left(\frac{1}{2}\right) \cdot \left(\frac{1}{2}\right) \cdot \left(\frac{1}{2}\right) \cdot \left(\frac{1}{2}\right) = \frac{1}{64}
\]
Answer: \(\frac{1}{64}\)
---
1. Simplify the fraction inside the parentheses:
\[
\frac{5}{10} = \frac{1}{2}
\]
2. Rewrite the negative exponent as a reciprocal:
\[
\left(\frac{1}{2}\right)^{-2} = \left(2\right)^2 = 4
\]
Answer: \(4\)
---
1. Simplify the fraction inside the parentheses:
\[
\frac{6}{8} = \frac{3}{4}
\]
2. Rewrite the negative exponent as a reciprocal:
\[
\left(\frac{3}{4}\right)^{-3} = \left(\frac{4}{3}\right)^3
\]
3. Raise \(\frac{4}{3}\) to the power of 3:
\[
\left(\frac{4}{3}\right)^3 = \left(\frac{4}{3}\right) \cdot \left(\frac{4}{3}\right) \cdot \left(\frac{4}{3}\right) = \frac{64}{27}
\]
Answer: \(\frac{64}{27}\)
---
1. Simplify the fraction inside the parentheses:
\[
\frac{2}{6} = \frac{1}{3}
\]
2. Square the simplified fraction:
\[
\left(\frac{1}{3}\right)^2 = \left(\frac{1}{3}\right) \cdot \left(\frac{1}{3}\right) = \frac{1}{9}
\]
Answer: \(\frac{1}{9}\)
---
1. Raise \(-\frac{1}{2}\) to the power of 8 (an even power makes the result positive):
\[
\left(-\frac{1}{2}\right)^8 = \left(\frac{1}{2}\right)^8 = \left(\frac{1}{2}\right) \cdot \left(\frac{1}{2}\right) \cdot \left(\frac{1}{2}\right) \cdot \left(\frac{1}{2}\right) \cdot \left(\frac{1}{2}\right) \cdot \left(\frac{1}{2}\right) \cdot \left(\frac{1}{2}\right) \cdot \left(\frac{1}{2}\right) = \frac{1}{256}
\]
Answer: \(\frac{1}{256}\)
---
1. Raise \(-\frac{2}{3}\) to the power of 5:
\[
\left(-\frac{2}{3}\right)^5 = \left(-\frac{2}{3}\right) \cdot \left(-\frac{2}{3}\right) \cdot \left(-\frac{2}{3}\right) \cdot \left(-\frac{2}{3}\right) \cdot \left(-\frac{2}{3}\right) = -\frac{32}{243}
\]
Answer: \(-\frac{32}{243}\)
---
1. Rewrite the negative exponent as a reciprocal:
\[
\left(-\frac{1}{2}\right)^{-7} = \left(-2\right)^7
\]
2. Raise \(-2\) to the power of 7:
\[
(-2)^7 = -128
\]
Answer: \(-128\)
---
1. Simplify the fraction inside the parentheses:
\[
\frac{2}{4} = \frac{1}{2}
\]
2. Square the simplified fraction:
\[
\left(\frac{1}{2}\right)^2 = \left(\frac{1}{2}\right) \cdot \left(\frac{1}{2}\right) = \frac{1}{4}
\]
Answer: \(\frac{1}{4}\)
---
1. Simplify the fraction inside the parentheses:
\[
\frac{2}{12} = \frac{1}{6}
\]
2. Square the simplified fraction:
\[
\left(\frac{1}{6}\right)^2 = \left(\frac{1}{6}\right) \cdot \left(\frac{1}{6}\right) = \frac{1}{36}
\]
Answer: \(\frac{1}{36}\)
---
1. Square the fraction:
\[
\left(-\frac{2}{9}\right)^2 = \left(-\frac{2}{9}\right) \cdot \left(-\frac{2}{9}\right) = \frac{4}{81}
\]
Answer: \(\frac{4}{81}\)
---
1. Raise \(\frac{1}{2}\) to the power of 6:
\[
\left(\frac{1}{2}\right)^6 = \left(\frac{1}{2}\right) \cdot \left(\frac{1}{2}\right) \cdot \left(\frac{1}{2}\right) \cdot \left(\frac{1}{2}\right) \cdot \left(\frac{1}{2}\right) \cdot \left(\frac{1}{2}\right) = \frac{1}{64}
\]
Answer: \(\frac{1}{64}\)
---
1. Raise \(\frac{1}{3}\) to the power of 4:
\[
\left(\frac{1}{3}\right)^4 = \left(\frac{1}{3}\right) \cdot \left(\frac{1}{3}\right) \cdot \left(\frac{1}{3}\right) \cdot \left(\frac{1}{3}\right) = \frac{1}{81}
\]
Answer: \(\frac{1}{81}\)
---
1. Simplify the fraction inside the parentheses:
\[
-\frac{3}{6} = -\frac{1}{2}
\]
2. Rewrite the negative exponent as a reciprocal:
\[
\left(-\frac{1}{2}\right)^{-2} = \left(-2\right)^2 = 4
\]
Answer: \(4\)
---
\[
\boxed{
\begin{array}{ll}
1. & \frac{1}{16} \\
2. & \frac{9}{16} \\
3. & \frac{1}{27} \\
4. & \frac{16}{25} \\
5. & -\frac{243}{32} \\
6. & \frac{1}{4} \\
7. & -\frac{8}{343} \\
8. & \frac{1}{64} \\
9. & 4 \\
10. & \frac{64}{27} \\
11. & \frac{1}{9} \\
12. & \frac{1}{256} \\
13. & -\frac{32}{243} \\
14. & -128 \\
15. & \frac{1}{4} \\
16. & \frac{1}{36} \\
17. & \frac{4}{81} \\
18. & \frac{1}{64} \\
19. & \frac{1}{81} \\
20. & 4 \\
\end{array}
}
\]
Problem 1: \(\left(-\frac{3}{12}\right)^2\)
1. Simplify the fraction inside the parentheses:
\[
-\frac{3}{12} = -\frac{1}{4}
\]
2. Square the simplified fraction:
\[
\left(-\frac{1}{4}\right)^2 = \left(-\frac{1}{4}\right) \cdot \left(-\frac{1}{4}\right) = \frac{1}{16}
\]
Answer: \(\frac{1}{16}\)
---
Problem 2: \(\left(-\frac{3}{4}\right)^2\)
1. Square the fraction:
\[
\left(-\frac{3}{4}\right)^2 = \left(-\frac{3}{4}\right) \cdot \left(-\frac{3}{4}\right) = \frac{9}{16}
\]
Answer: \(\frac{9}{16}\)
---
Problem 3: \(\left(\frac{1}{3}\right)^3\)
1. Cube the fraction:
\[
\left(\frac{1}{3}\right)^3 = \left(\frac{1}{3}\right) \cdot \left(\frac{1}{3}\right) \cdot \left(\frac{1}{3}\right) = \frac{1}{27}
\]
Answer: \(\frac{1}{27}\)
---
Problem 4: \(\left(-\frac{4}{5}\right)^2\)
1. Square the fraction:
\[
\left(-\frac{4}{5}\right)^2 = \left(-\frac{4}{5}\right) \cdot \left(-\frac{4}{5}\right) = \frac{16}{25}
\]
Answer: \(\frac{16}{25}\)
---
Problem 5: \(\left(-\frac{2}{3}\right)^{-5}\)
1. Rewrite the negative exponent as a reciprocal:
\[
\left(-\frac{2}{3}\right)^{-5} = \left(\frac{3}{-2}\right)^5 = \left(-\frac{3}{2}\right)^5
\]
2. Raise \(-\frac{3}{2}\) to the power of 5:
\[
\left(-\frac{3}{2}\right)^5 = \left(-\frac{3}{2}\right) \cdot \left(-\frac{3}{2}\right) \cdot \left(-\frac{3}{2}\right) \cdot \left(-\frac{3}{2}\right) \cdot \left(-\frac{3}{2}\right) = -\frac{243}{32}
\]
Answer: \(-\frac{243}{32}\)
---
Problem 6: \(\left(-\frac{5}{10}\right)^2\)
1. Simplify the fraction inside the parentheses:
\[
-\frac{5}{10} = -\frac{1}{2}
\]
2. Square the simplified fraction:
\[
\left(-\frac{1}{2}\right)^2 = \left(-\frac{1}{2}\right) \cdot \left(-\frac{1}{2}\right) = \frac{1}{4}
\]
Answer: \(\frac{1}{4}\)
---
Problem 7: \(\left(-\frac{2}{7}\right)^3\)
1. Cube the fraction:
\[
\left(-\frac{2}{7}\right)^3 = \left(-\frac{2}{7}\right) \cdot \left(-\frac{2}{7}\right) \cdot \left(-\frac{2}{7}\right) = -\frac{8}{343}
\]
Answer: \(-\frac{8}{343}\)
---
Problem 8: \(\left(\frac{1}{2}\right)^6\)
1. Raise \(\frac{1}{2}\) to the power of 6:
\[
\left(\frac{1}{2}\right)^6 = \left(\frac{1}{2}\right) \cdot \left(\frac{1}{2}\right) \cdot \left(\frac{1}{2}\right) \cdot \left(\frac{1}{2}\right) \cdot \left(\frac{1}{2}\right) \cdot \left(\frac{1}{2}\right) = \frac{1}{64}
\]
Answer: \(\frac{1}{64}\)
---
Problem 9: \(\left(\frac{5}{10}\right)^{-2}\)
1. Simplify the fraction inside the parentheses:
\[
\frac{5}{10} = \frac{1}{2}
\]
2. Rewrite the negative exponent as a reciprocal:
\[
\left(\frac{1}{2}\right)^{-2} = \left(2\right)^2 = 4
\]
Answer: \(4\)
---
Problem 10: \(\left(\frac{6}{8}\right)^{-3}\)
1. Simplify the fraction inside the parentheses:
\[
\frac{6}{8} = \frac{3}{4}
\]
2. Rewrite the negative exponent as a reciprocal:
\[
\left(\frac{3}{4}\right)^{-3} = \left(\frac{4}{3}\right)^3
\]
3. Raise \(\frac{4}{3}\) to the power of 3:
\[
\left(\frac{4}{3}\right)^3 = \left(\frac{4}{3}\right) \cdot \left(\frac{4}{3}\right) \cdot \left(\frac{4}{3}\right) = \frac{64}{27}
\]
Answer: \(\frac{64}{27}\)
---
Problem 11: \(\left(\frac{2}{6}\right)^2\)
1. Simplify the fraction inside the parentheses:
\[
\frac{2}{6} = \frac{1}{3}
\]
2. Square the simplified fraction:
\[
\left(\frac{1}{3}\right)^2 = \left(\frac{1}{3}\right) \cdot \left(\frac{1}{3}\right) = \frac{1}{9}
\]
Answer: \(\frac{1}{9}\)
---
Problem 12: \(\left(-\frac{1}{2}\right)^8\)
1. Raise \(-\frac{1}{2}\) to the power of 8 (an even power makes the result positive):
\[
\left(-\frac{1}{2}\right)^8 = \left(\frac{1}{2}\right)^8 = \left(\frac{1}{2}\right) \cdot \left(\frac{1}{2}\right) \cdot \left(\frac{1}{2}\right) \cdot \left(\frac{1}{2}\right) \cdot \left(\frac{1}{2}\right) \cdot \left(\frac{1}{2}\right) \cdot \left(\frac{1}{2}\right) \cdot \left(\frac{1}{2}\right) = \frac{1}{256}
\]
Answer: \(\frac{1}{256}\)
---
Problem 13: \(\left(-\frac{2}{3}\right)^5\)
1. Raise \(-\frac{2}{3}\) to the power of 5:
\[
\left(-\frac{2}{3}\right)^5 = \left(-\frac{2}{3}\right) \cdot \left(-\frac{2}{3}\right) \cdot \left(-\frac{2}{3}\right) \cdot \left(-\frac{2}{3}\right) \cdot \left(-\frac{2}{3}\right) = -\frac{32}{243}
\]
Answer: \(-\frac{32}{243}\)
---
Problem 14: \(\left(-\frac{1}{2}\right)^{-7}\)
1. Rewrite the negative exponent as a reciprocal:
\[
\left(-\frac{1}{2}\right)^{-7} = \left(-2\right)^7
\]
2. Raise \(-2\) to the power of 7:
\[
(-2)^7 = -128
\]
Answer: \(-128\)
---
Problem 15: \(\left(\frac{2}{4}\right)^2\)
1. Simplify the fraction inside the parentheses:
\[
\frac{2}{4} = \frac{1}{2}
\]
2. Square the simplified fraction:
\[
\left(\frac{1}{2}\right)^2 = \left(\frac{1}{2}\right) \cdot \left(\frac{1}{2}\right) = \frac{1}{4}
\]
Answer: \(\frac{1}{4}\)
---
Problem 16: \(\left(\frac{2}{12}\right)^2\)
1. Simplify the fraction inside the parentheses:
\[
\frac{2}{12} = \frac{1}{6}
\]
2. Square the simplified fraction:
\[
\left(\frac{1}{6}\right)^2 = \left(\frac{1}{6}\right) \cdot \left(\frac{1}{6}\right) = \frac{1}{36}
\]
Answer: \(\frac{1}{36}\)
---
Problem 17: \(\left(-\frac{2}{9}\right)^2\)
1. Square the fraction:
\[
\left(-\frac{2}{9}\right)^2 = \left(-\frac{2}{9}\right) \cdot \left(-\frac{2}{9}\right) = \frac{4}{81}
\]
Answer: \(\frac{4}{81}\)
---
Problem 18: \(\left(\frac{1}{2}\right)^6\)
1. Raise \(\frac{1}{2}\) to the power of 6:
\[
\left(\frac{1}{2}\right)^6 = \left(\frac{1}{2}\right) \cdot \left(\frac{1}{2}\right) \cdot \left(\frac{1}{2}\right) \cdot \left(\frac{1}{2}\right) \cdot \left(\frac{1}{2}\right) \cdot \left(\frac{1}{2}\right) = \frac{1}{64}
\]
Answer: \(\frac{1}{64}\)
---
Problem 19: \(\left(\frac{1}{3}\right)^4\)
1. Raise \(\frac{1}{3}\) to the power of 4:
\[
\left(\frac{1}{3}\right)^4 = \left(\frac{1}{3}\right) \cdot \left(\frac{1}{3}\right) \cdot \left(\frac{1}{3}\right) \cdot \left(\frac{1}{3}\right) = \frac{1}{81}
\]
Answer: \(\frac{1}{81}\)
---
Problem 20: \(\left(-\frac{3}{6}\right)^{-2}\)
1. Simplify the fraction inside the parentheses:
\[
-\frac{3}{6} = -\frac{1}{2}
\]
2. Rewrite the negative exponent as a reciprocal:
\[
\left(-\frac{1}{2}\right)^{-2} = \left(-2\right)^2 = 4
\]
Answer: \(4\)
---
Final Answers:
\[
\boxed{
\begin{array}{ll}
1. & \frac{1}{16} \\
2. & \frac{9}{16} \\
3. & \frac{1}{27} \\
4. & \frac{16}{25} \\
5. & -\frac{243}{32} \\
6. & \frac{1}{4} \\
7. & -\frac{8}{343} \\
8. & \frac{1}{64} \\
9. & 4 \\
10. & \frac{64}{27} \\
11. & \frac{1}{9} \\
12. & \frac{1}{256} \\
13. & -\frac{32}{243} \\
14. & -128 \\
15. & \frac{1}{4} \\
16. & \frac{1}{36} \\
17. & \frac{4}{81} \\
18. & \frac{1}{64} \\
19. & \frac{1}{81} \\
20. & 4 \\
\end{array}
}
\]
Parent Tip: Review the logic above to help your child master the concept of simplifying exponents worksheet pdf.
![How To Use Simplifying Exponents Worksheets PDFs] Brighterly.com](/images2/simplifying-exponents-worksheet-pdf/simplifying-exponents-worksheet-pdf-8.webp)
