Question

radicals and rational exponents
write each expression in radical form.

1. $7^{\frac{1}{2}}$
2. $4^{\frac{4}{3}}$
3. $2^{\frac{5}{3}}$
4. $7^{\frac{4}{3}}$
5. $6^{\frac{3}{2}}$
6. $2^{\frac{1}{6}}$

write each expression in exponential form.

1. $\left(\sqrt{10}\

ight)^3$

1. $\sqrt6{2}$
2. $\left(\sqrt4{2}\

ight)^5$

1. $\left(\sqrt4{5}\

ight)^5$

1. $\sqrt5{2}$
2. $\sqrt6{10}$

write each expression in radical form.

1. $\left(5x\

ight)^{-\frac{5}{4}}$

1. $\left(5x\

ight)^{-\frac{1}{2}}$

1. $\left(10n\

ight)^{\frac{3}{2}}$

1. $a^{\frac{6}{5}}$

Explanation:

Step1: Use rule $a^{\frac{m}{n}}=\sqrt[n]{a^m}$

For expressions with positive rational exponents, convert to radical form by placing the denominator as the root index and numerator as the power of the base.

Step2: Use rule $a^{-\frac{m}{n}}=\frac{1}{a^{\frac{m}{n}}}=\frac{1}{\sqrt[n]{a^m}}$

For negative rational exponents, rewrite as the reciprocal of the positive exponent form first, then convert to radical.

Step3: Use rule $(\sqrt[n]{a})^m=a^{\frac{m}{n}}$

For radical expressions raised to a power, convert to exponential form by using the root index as the denominator and the power as the numerator of the exponent.

Step4: Use rule $\sqrt[n]{a}=a^{\frac{1}{n}}$

For simple radicals (no outer power), convert to exponential form with 1 as the numerator of the exponent.

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Part 1: Convert to Radical Form

1) Convert $7^{\frac{1}{2}}$

$\sqrt{7}$

2) Convert $4^{\frac{4}{3}}$

$\sqrt[3]{4^4}$ or $\sqrt[3]{256}$

3) Convert $2^{\frac{5}{3}}$

$\sqrt[3]{2^5}$ or $\sqrt[3]{32}$

4) Convert $7^{\frac{4}{3}}$

$\sqrt[3]{7^4}$ or $\sqrt[3]{2401}$

5) Convert $6^{\frac{3}{2}}$

$\sqrt{6^3}$ or $\sqrt{216}$

6) Convert $2^{\frac{1}{6}}$

$\sqrt[6]{2}$

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Part 2: Convert to Exponential Form

7) Convert $(\sqrt{10})^3$

$10^{\frac{3}{2}}$

8) Convert $\sqrt[6]{2}$

$2^{\frac{1}{6}}$

9) Convert $(\sqrt[4]{2})^5$

$2^{\frac{5}{4}}$

10) Convert $(\sqrt[4]{5})^5$

$5^{\frac{5}{4}}$

11) Convert $\sqrt[3]{2}$

$2^{\frac{1}{3}}$

12) Convert $\sqrt[6]{10}$

$10^{\frac{1}{6}}$

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Part 3: Convert to Radical Form

13) Convert $(5x)^{-\frac{5}{4}}$

$\frac{1}{\sqrt[4]{(5x)^5}}$

14) Convert $(5x)^{-\frac{1}{2}}$

$\frac{1}{\sqrt{5x}}$

15) Convert $(10n)^{\frac{3}{2}}$

$\sqrt{(10n)^3}$ or $\sqrt{1000n^3}$

16) Convert $a^{\frac{6}{5}}$

$\sqrt[5]{a^6}$

Answer:

Radical Form (1-6, 13-16)

1. $\sqrt{7}$
2. $\sqrt[3]{4^4}$
3. $\sqrt[3]{2^5}$
4. $\sqrt[3]{7^4}$
5. $\sqrt{6^3}$
6. $\sqrt[6]{2}$
7. $\frac{1}{\sqrt[4]{(5x)^5}}$
8. $\frac{1}{\sqrt{5x}}$
9. $\sqrt{(10n)^3}$
10. $\sqrt[5]{a^6}$

Exponential Form (7-12)

1. $10^{\frac{3}{2}}$
2. $2^{\frac{1}{6}}$
3. $2^{\frac{5}{4}}$
4. $5^{\frac{5}{4}}$
5. $2^{\frac{1}{3}}$
6. $10^{\frac{1}{6}}$