Question

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date
lesson 6.1 practice
for use with pages 414-419
rewrite the expression using rational exponent notation.

1. $sqrt3{7}$
2. $(sqrt3{6})^2$
3. $(sqrt3{14})^4$
4. $(sqrt7{-21})^3$
5. $(sqrt9{11})^7$
6. $(sqrt9{-2})^4$

rewrite the expression using radical notation.

1. $17^{1/3}$
2. $44^{1/6}$
3. $33^{2/3}$
4. $9^{5/3}$
5. $(-28)^{7/5}$
6. $39^{4/7}$

evaluate the expression without using a calculator.

1. $(sqrt3{8})^2$
2. $(sqrt4{16})^3$
3. $(sqrt4{81})^4$
4. $36^{3/2}$
5. $4^{5/2}$
6. $27^{2/3}$
7. $125^{4/3}$
8. $(-8)^{1/3}$
9. $(-32)^{3/5}$

Explanation:

Step1: Convert radical to rational exponent

For $\sqrt[n]{a} = a^{\frac{1}{n}}$ and $(\sqrt[n]{a})^m = a^{\frac{m}{n}}$

1. $\sqrt[3]{7} = 7^{\frac{1}{3}}$
2. $(\sqrt[3]{6})^2 = 6^{\frac{2}{3}}$
3. $(\sqrt[9]{14})^4 = 14^{\frac{4}{9}}$
4. $(\sqrt[7]{-21})^3 = (-21)^{\frac{3}{7}}$
5. $(\sqrt[8]{11})^7 = 11^{\frac{7}{8}}$
6. $(\sqrt[9]{-2})^4 = (-2)^{\frac{4}{9}}$

Step2: Convert rational exponent to radical

For $a^{\frac{1}{n}} = \sqrt[n]{a}$ and $a^{\frac{m}{n}} = (\sqrt[n]{a})^m$

1. $17^{\frac{1}{3}} = \sqrt[3]{17}$
2. $44^{\frac{1}{6}} = \sqrt[6]{44}$
3. $33^{\frac{2}{3}} = (\sqrt[3]{33})^2$
4. $9^{\frac{5}{3}} = (\sqrt[3]{9})^5$
5. $(-28)^{\frac{7}{5}} = (\sqrt[5]{-28})^7$
6. $39^{\frac{4}{7}} = (\sqrt[7]{39})^4$

Step3: Evaluate radical expressions

Simplify roots first, then raise to power

1. $\sqrt[3]{8}=2$, so $(\sqrt[3]{8})^2 = 2^2 = 4$
2. $\sqrt[4]{16}=2$, so $(\sqrt[4]{16})^3 = 2^3 = 8$
3. $(\sqrt[4]{81})^4 = 81$ (root and exponent cancel)
4. $36^{\frac{3}{2}} = (\sqrt{36})^3 = 6^3 = 216$
5. $4^{\frac{5}{2}} = (\sqrt{4})^5 = 2^5 = 32$
6. $27^{\frac{2}{3}} = (\sqrt[3]{27})^2 = 3^2 = 9$
7. $125^{\frac{4}{3}} = (\sqrt[3]{125})^4 = 5^4 = 625$
8. $(-8)^{\frac{1}{3}} = \sqrt[3]{-8} = -2$
9. $(-32)^{\frac{3}{5}} = (\sqrt[5]{-32})^3 = (-2)^3 = -8$

Answer:

1. $7^{\frac{1}{3}}$
2. $6^{\frac{2}{3}}$
3. $14^{\frac{4}{9}}$
4. $(-21)^{\frac{3}{7}}$
5. $11^{\frac{7}{8}}$
6. $(-2)^{\frac{4}{9}}$
7. $\sqrt[3]{17}$
8. $\sqrt[6]{44}$
9. $(\sqrt[3]{33})^2$
10. $(\sqrt[3]{9})^5$
11. $(\sqrt[5]{-28})^7$
12. $(\sqrt[7]{39})^4$
13. $4$
14. $8$
15. $81$
16. $216$
17. $32$
18. $9$
19. $625$
20. $-2$
21. $-8$