# Honors Algebra 2: Extra Problems on Exponents and Radicals

• Class: Honors Algebra 2
• Author: Peter Atlas
• Text: Algebra and Trigonometry: Structure and Method, Brown

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1. Simplify: $$\displaystyle \left( x^{-1} - y^{-1} \right)^{-1}$$
2. Solution

$$\displaystyle \frac{xy}{y-x}$$

3. Simplify: $$\displaystyle \frac{x^{-1} + y^{-1}}{x^{-2}-y^{-2}}$$
4. Solution

$$\displaystyle \frac{xy}{y-x}$$

5. Simplify: $$\displaystyle \sqrt[3]{x^2} \cdot \sqrt[5]{x^3} \cdot \sqrt[6]{x}$$
6. Solution

$$\displaystyle x\sqrt[30]{x^{13}}$$

7. Simplify: $$\displaystyle \sqrt{a \cdot \sqrt[3]{a}}$$
8. Solution

$$\displaystyle \sqrt[3]{a^2}$$

9. Simplify: $$\displaystyle \left( \frac{27^{\frac{4}{3}} - 27^0}{ \left( 3^2 + 4^2 \right)^{\frac{1}{2}}} \right)^ \frac{3}{4}$$
10. Solution

$$\displaystyle 8$$

11. Simplify: $$\displaystyle \left( \frac{a^{-2}b^{\frac{2}{3}}}{b^{-\frac{1}{2}}} \right)^{-4}$$
12. Solution

$$\displaystyle \frac{a^8}{b^{\frac{14}{3}}}$$

13. Simplify: $$\displaystyle \frac{3^{-1}}{3^{-2}+3^{-3}}$$
14. Solution

$$\displaystyle \frac{9}{4}$$

15. Solve: $$\displaystyle 4^{x+1} = \left( \frac{1}{2} \right)^{-3x}$$
16. Solution

$$\displaystyle 2$$

17. Solve: $$\displaystyle 2^{4x}\cdot 4^{x - 3} = 64^{x - 1}$$
18. Solution

$$\displaystyle x \in \mathbb{R}$$

19. Solve: $$\displaystyle 3 \left( x - 4 \right)^{\frac{1}{2}} = 15$$
20. Solution

$$\displaystyle 29$$

21. Solve: $$\displaystyle x^{\frac{2}{3}} - 3x^{\frac{1}{3}} = -2$$
22. Solution

$$\displaystyle x = \{ 1, 8 \}$$