# Honors Algebra 2: Extra Practice on Rational Expressions

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

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Rewrite using exponential notation:

1. $$\displaystyle \frac{\sqrt[4]{x^3y^5}}{2a^7\sqrt{b}}$$
2. Solution

$$\displaystyle x^{\frac{3}{4}}y^{\frac{5}{4}}2^{-1}a^{-7}b^{-\frac{1}{2}}$$

3. $$\displaystyle \sqrt[12]{\sqrt[5]{x} + y\sqrt{y}}$$
4. Solution

$$\displaystyle \left(x^{\frac{1}{5}} + y \cdot y^{\frac{1}{2}}\right)^{\frac{1}{12}}$$

Rewrite using radical notation and no negative exponents:

1. $$\displaystyle ab^{\frac{2}{3}}(3c)^{-5/7}$$
2. Solution

$$\displaystyle \frac{a\sqrt[3]{b^2}}{\sqrt[7]{\left(3c\right)^5}}$$

3. $$\displaystyle 5^{\frac{1}{2}}x^{-\frac{1}{2}}$$
4. Solution

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

5. $$\displaystyle \left(3y\right)^{-\frac{2}{5}}$$
6. Solution

$$\displaystyle \frac{1}{\sqrt[5]{9y^2}}$$

7. $$\displaystyle 3y^{-\frac{2}{5}}$$
8. Solution

$$\displaystyle \frac{3}{\sqrt[5]{y^2}}$$

Simplify the following:

1. $$\displaystyle 2n^{\frac{1}{3}}\left(n^{\frac{2}{3}} + n^{-\frac{1}{3}}\right)$$
2. Solution

$$\displaystyle 2n + 2$$

3. $$\displaystyle \frac{x^{\frac{1}{2}} - 2x^{-\frac{1}{2}}}{x^{-\frac{1}{2}}}$$
4. Solution

$$\displaystyle x - 2$$

5. $$\displaystyle \frac{y^{-\frac{1}{3}} - 3y^{\frac{2}{3}}}{y^{-\frac{4}{3}}}$$
6. Solution

$$\displaystyle y - 3y^{2}$$

7. $$\displaystyle \frac{2n^{\frac{1}{3}}\left(3n^{\frac{1}{3}} - 4n^{\frac{4}{3}}\right)}{2n^{-\frac{1}{3}}}$$
8. Solution

$$\displaystyle 3n - 4n^{2}$$

9. $$\displaystyle \frac{\left(\sqrt{2x}\right)^{5}}{\left(x\sqrt{2}\right)^{9}}$$
10. Solution

$$\displaystyle \frac{1}{4x^6\sqrt{x}}$$

Factor the following:

1. $$\displaystyle a^{\frac{3}{2}}b^{\frac{1}{2}} - a^{\frac{1}{2}}b^{\frac{3}{2}}$$
2. Solution

$$\displaystyle a^{\frac{1}{2}} b^{\frac{1}{2}} \left(a - b\right)$$

3. $$\displaystyle \left(x - 1\right)^{\frac{1}{2}} - x\left(x - 1\right)^{-\frac{1}{2}}$$
4. Solution

$$\displaystyle -\left(x - 1\right)^{-\frac{1}{2}}$$

5. $$\displaystyle \left(x^{2} + 1\right)^{\frac{3}{2}} - x^{2} \left(x^{2} + 1\right)^{\frac{1}{2}}$$
6. Solution

$$\displaystyle \left(x^{2} +1\right)^{\frac{1}{2}}$$

7. $$\displaystyle \left(2x + 1\right)^{\frac{2}{3}} - 4\left(2x + 1\right)^{-\frac{1}{3}}$$
8. Solution

$$\displaystyle \left(2x + 1\right)^{-\frac{1}{3}} \left(2x - 3\right)$$

Simplify the following:

1. $$\displaystyle 5\sqrt{4x^6}$$
2. Solution

$$\displaystyle 10 \left| x^3 \right|$$

3. $$\displaystyle \sqrt{189x^5}$$
4. Solution

$$\displaystyle 3x^2\sqrt{21x}$$

5. $$\displaystyle \sqrt[4]{\left(-81\right)^{-2}}$$
6. Solution

$$\displaystyle \frac{1}{9}$$

7. $$\displaystyle \sqrt[6]{8} \cdot \sqrt[8]{81}$$
8. Solution

$$\displaystyle \sqrt{6}$$

9. $$\displaystyle \frac{7}{\sqrt{10} - \sqrt{15}}$$
10. Solution

$$\displaystyle - \frac{7\sqrt{10} + 7\sqrt{15}}{5}$$

11. $$\displaystyle \frac{\sqrt{x + y}}{5 - \sqrt{x + y}}$$
12. Solution

$$\displaystyle \frac{5\sqrt{x + y} + x + y}{25 - x - y}$$

13. $$\displaystyle \sqrt[4]{\frac{7}{25a^3}}; a > 0$$
14. Solution

$$\displaystyle \frac{\sqrt[4]{175a}}{5a}$$

15. $$\displaystyle 5\sqrt{12x} - 6\sqrt{48x} + \sqrt{75x}$$
16. Solution

$$\displaystyle -9\sqrt{3x}$$