Rational Exponents Worksheets - Math Monks - Free Printable

Problem Analysis

The task involves working with nth roots and rational exponents. We need to:
1. Convert expressions from exponential form to radical form.
2. Convert expressions from radical form to exponential form.
3. Simplify given expressions.

Let's solve each part step by step.

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Part 1: Write each expression in radical form

#### Expression 1: \( 3^{\frac{5}{2}} \)
- The general rule is: \( a^{\frac{m}{n}} = \sqrt[n]{a^m} \).
- Here, \( a = 3 \), \( m = 5 \), and \( n = 2 \).
- Therefore, \( 3^{\frac{5}{2}} = \sqrt[2]{3^5} = \sqrt{3^5} \).

Answer: \( \sqrt{3^5} \)

#### Expression 2: \( 10^{\frac{4}{3}} \)
- Using the same rule: \( a^{\frac{m}{n}} = \sqrt[n]{a^m} \).
- Here, \( a = 10 \), \( m = 4 \), and \( n = 3 \).
- Therefore, \( 10^{\frac{4}{3}} = \sqrt[3]{10^4} \).

Answer: \( \sqrt[3]{10^4} \)

#### Expression 3: \( (7b)^{\frac{2}{3}} \)
- The base is \( 7b \), so we apply the rule directly: \( (7b)^{\frac{2}{3}} = \sqrt[3]{(7b)^2} \).

Answer: \( \sqrt[3]{(7b)^2} \)

#### Expression 4: \( (7m)^{-\frac{5}{2}} \)
- For negative exponents, \( a^{-\frac{m}{n}} = \frac{1}{\sqrt[n]{a^m}} \).
- Here, \( a = 7m \), \( m = 5 \), and \( n = 2 \).
- Therefore, \( (7m)^{-\frac{5}{2}} = \frac{1}{\sqrt[2]{(7m)^5}} = \frac{1}{\sqrt{(7m)^5}} \).

Answer: \( \frac{1}{\sqrt{(7m)^5}} \)

#### Expression 5: \( (b^2)^{\frac{1}{3}} \)
- Using the rule \( (a^m)^n = a^{m \cdot n} \), we first simplify the exponent: \( (b^2)^{\frac{1}{3}} = b^{2 \cdot \frac{1}{3}} = b^{\frac{2}{3}} \).
- Now convert to radical form: \( b^{\frac{2}{3}} = \sqrt[3]{b^2} \).

Answer: \( \sqrt[3]{b^2} \)

#### Expression 6: \( (7n)^{\frac{5}{2}} \)
- Using the rule \( a^{\frac{m}{n}} = \sqrt[n]{a^m} \):
- Here, \( a = 7n \), \( m = 5 \), and \( n = 2 \).
- Therefore, \( (7n)^{\frac{5}{2}} = \sqrt[2]{(7n)^5} = \sqrt{(7n)^5} \).

Answer: \( \sqrt{(7n)^5} \)

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Part 2: Write each expression in exponential form

#### Expression 7: \( (\sqrt[4]{3p})^7 \)
- The fourth root can be written as an exponent: \( \sqrt[4]{3p} = (3p)^{\frac{1}{4}} \).
- Raising this to the 7th power: \( (\sqrt[4]{3p})^7 = \left( (3p)^{\frac{1}{4}} \right)^7 \).
- Using the exponent rule \( (a^m)^n = a^{m \cdot n} \): \( \left( (3p)^{\frac{1}{4}} \right)^7 = (3p)^{\frac{1}{4} \cdot 7} = (3p)^{\frac{7}{4}} \).

Answer: \( (3p)^{\frac{7}{4}} \)

#### Expression 8: \( \frac{1}{\sqrt[6]{10}} \)
- The sixth root can be written as an exponent: \( \sqrt[6]{10} = 10^{\frac{1}{6}} \).
- The reciprocal of this is: \( \frac{1}{\sqrt[6]{10}} = \frac{1}{10^{\frac{1}{6}}} = 10^{-\frac{1}{6}} \).

Answer: \( 10^{-\frac{1}{6}} \)

#### Expression 9: \( \frac{1}{(\sqrt[3]{3x})^4} \)
- The cube root can be written as an exponent: \( \sqrt[3]{3x} = (3x)^{\frac{1}{3}} \).
- Raising this to the 4th power: \( (\sqrt[3]{3x})^4 = \left( (3x)^{\frac{1}{3}} \right)^4 = (3x)^{\frac{1}{3} \cdot 4} = (3x)^{\frac{4}{3}} \).
- The reciprocal of this is: \( \frac{1}{(\sqrt[3]{3x})^4} = \frac{1}{(3x)^{\frac{4}{3}}} = (3x)^{-\frac{4}{3}} \).

Answer: \( (3x)^{-\frac{4}{3}} \)

#### Expression 10: \( (\sqrt[5]{10x})^6 \)
- The fifth root can be written as an exponent: \( \sqrt[5]{10x} = (10x)^{\frac{1}{5}} \).
- Raising this to the 6th power: \( (\sqrt[5]{10x})^6 = \left( (10x)^{\frac{1}{5}} \right)^6 = (10x)^{\frac{1}{5} \cdot 6} = (10x)^{\frac{6}{5}} \).

Answer: \( (10x)^{\frac{6}{5}} \)

#### Expression 11: \( (\sqrt[7]{10})^3 \)
- The seventh root can be written as an exponent: \( \sqrt[7]{10} = 10^{\frac{1}{7}} \).
- Raising this to the 3rd power: \( (\sqrt[7]{10})^3 = \left( 10^{\frac{1}{7}} \right)^3 = 10^{\frac{1}{7} \cdot 3} = 10^{\frac{3}{7}} \).

Answer: \( 10^{\frac{3}{7}} \)

#### Expression 12: \( \frac{1}{(\sqrt[3]{x})^2} \)
- The cube root can be written as an exponent: \( \sqrt[3]{x} = x^{\frac{1}{3}} \).
- Raising this to the 2nd power: \( (\sqrt[3]{x})^2 = \left( x^{\frac{1}{3}} \right)^2 = x^{\frac{1}{3} \cdot 2} = x^{\frac{2}{3}} \).
- The reciprocal of this is: \( \frac{1}{(\sqrt[3]{x})^2} = \frac{1}{x^{\frac{2}{3}}} = x^{-\frac{2}{3}} \).

Answer: \( x^{-\frac{2}{3}} \)

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Part 3: Simplify

#### Expression 13: \( \sqrt[3]{-250} \)
- Factorize \( -250 \): \( -250 = -1 \cdot 125 \cdot 2 \).
- Note that \( 125 = 5^3 \), so \( \sqrt[3]{-250} = \sqrt[3]{-1 \cdot 5^3 \cdot 2} \).
- Using the property of cube roots: \( \sqrt[3]{-1 \cdot 5^3 \cdot 2} = \sqrt[3]{-1} \cdot \sqrt[3]{5^3} \cdot \sqrt[3]{2} = -1 \cdot 5 \cdot \sqrt[3]{2} = -5\sqrt[3]{2} \).

Answer: \( -5\sqrt[3]{2} \)

#### Expression 14: \( \sqrt[3]{375} \)
- Factorize \( 375 \): \( 375 = 125 \cdot 3 \).
- Note that \( 125 = 5^3 \), so \( \sqrt[3]{375} = \sqrt[3]{5^3 \cdot 3} \).
- Using the property of cube roots: \( \sqrt[3]{5^3 \cdot 3} = \sqrt[3]{5^3} \cdot \sqrt[3]{3} = 5\sqrt[3]{3} \).

Answer: \( 5\sqrt[3]{3} \)

#### Expression 15: \( \sqrt[6]{384x^6y^5} \)
- Factorize \( 384 \): \( 384 = 64 \cdot 6 = 4^3 \cdot 6 \).
- Rewrite the expression: \( \sqrt[6]{384x^6y^5} = \sqrt[6]{4^3 \cdot 6 \cdot x^6 \cdot y^5} \).
- Using the property of sixth roots: \( \sqrt[6]{4^3 \cdot 6 \cdot x^6 \cdot y^5} = \sqrt[6]{4^3} \cdot \sqrt[6]{6} \cdot \sqrt[6]{x^6} \cdot \sqrt[6]{y^5} \).
- Simplify each term:
- \( \sqrt[6]{4^3} = (4^3)^{\frac{1}{6}} = 4^{\frac{3}{6}} = 4^{\frac{1}{2}} = \sqrt{4} = 2 \).
- \( \sqrt[6]{x^6} = x \).
- \( \sqrt[6]{y^5} = y^{\frac{5}{6}} \).
- \( \sqrt[6]{6} \) remains as it is.
- Combine the results: \( 2 \cdot x \cdot \sqrt[6]{6} \cdot y^{\frac{5}{6}} = 2x \sqrt[6]{6y^5} \).

Answer: \( 2x \sqrt[6]{6y^5} \)

#### Expression 16: \( (125)^{\frac{4}{3}} \)
- Note that \( 125 = 5^3 \), so \( (125)^{\frac{4}{3}} = (5^3)^{\frac{4}{3}} \).
- Using the exponent rule \( (a^m)^n = a^{m \cdot n} \): \( (5^3)^{\frac{4}{3}} = 5^{3 \cdot \frac{4}{3}} = 5^4 \).
- Calculate \( 5^4 \): \( 5^4 = 625 \).

Answer: \( 625 \)

#### Expression 17: \( (216)^{\frac{2}{3}} \)
- Note that \( 216 = 6^3 \), so \( (216)^{\frac{2}{3}} = (6^3)^{\frac{2}{3}} \).
- Using the exponent rule \( (a^m)^n = a^{m \cdot n} \): \( (6^3)^{\frac{2}{3}} = 6^{3 \cdot \frac{2}{3}} = 6^2 \).
- Calculate \( 6^2 \): \( 6^2 = 36 \).

Answer: \( 36 \)

#### Expression 18: \( (32)^{\frac{3}{5}} \)
- Note that \( 32 = 2^5 \), so \( (32)^{\frac{3}{5}} = (2^5)^{\frac{3}{5}} \).
- Using the exponent rule \( (a^m)^n = a^{m \cdot n} \): \( (2^5)^{\frac{3}{5}} = 2^{5 \cdot \frac{3}{5}} = 2^3 \).
- Calculate \( 2^3 \): \( 2^3 = 8 \).

Answer: \( 8 \)

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Final Answers

1. \( \sqrt{3^5} \)
2. \( \sqrt[3]{10^4} \)
3. \( \sqrt[3]{(7b)^2} \)
4. \( \frac{1}{\sqrt{(7m)^5}} \)
5. \( \sqrt[3]{b^2} \)
6. \( \sqrt{(7n)^5} \)
7. \( (3p)^{\frac{7}{4}} \)
8. \( 10^{-\frac{1}{6}} \)
9. \( (3x)^{-\frac{4}{3}} \)
10. \( (10x)^{\frac{6}{5}} \)
11. \( 10^{\frac{3}{7}} \)
12. \( x^{-\frac{2}{3}} \)
13. \( -5\sqrt[3]{2} \)
14. \( 5\sqrt[3]{3} \)
15. \( 2x \sqrt[6]{6y^5} \)
16. \( 625 \)
17. \( 36 \)
18. \( 8 \)

\boxed{
\begin{aligned}
&1. \sqrt{3^5} \\
&2. \sqrt[3]{10^4} \\
&3. \sqrt[3]{(7b)^2} \\
&4. \frac{1}{\sqrt{(7m)^5}} \\
&5. \sqrt[3]{b^2} \\
&6. \sqrt{(7n)^5} \\
&7. (3p)^{\frac{7}{4}} \\
&8. 10^{-\frac{1}{6}} \\
&9. (3x)^{-\frac{4}{3}} \\
&10. (10x)^{\frac{6}{5}} \\
&11. 10^{\frac{3}{7}} \\
&12. x^{-\frac{2}{3}} \\
&13. -5\sqrt[3]{2} \\
&14. 5\sqrt[3]{3} \\
&15. 2x \sqrt[6]{6y^5} \\
&16. 625 \\
&17. 36 \\
&18. 8
\end{aligned}
}