Review - Exponents, Radicals, Factoring, & Rational Expressions

1. Factor Completely:

1. $\displaystyle{16a^2+48a+36}$   $\ans{\displaystyle{4(2a+3)^2}}$

2. $\displaystyle{x^3-343}$   $\ans{\displaystyle{(x-7)(x^2+7x+49)}}$

3. $\displaystyle{m^5-9m^3n^2}$   $\ans{\displaystyle{m^3(m-3n)(m+3n)}}$

4. $\displaystyle{3x+9y+x^2+3xy}$   $\ans{\displaystyle{(3+x)(x+3y)}}$

5. $\displaystyle{a^8-8a^2b^6}$   $\ans{\displaystyle{a^2(a^2-2b^2)(a^4+2a^2b^2+4b^4)}}$

6. $\displaystyle{2xa^2+3a^2-18x-27}$   $\ans{\displaystyle{(2x+3)(a+3)(a-3)}}$

2. Rationalize the denominator:

1. $\displaystyle{\frac{7-\sqrt{x}}{7+\sqrt{x}}}$   $\ans{\displaystyle{\frac{49-14\sqrt{x}+x}{49-x}}}$

2. $\displaystyle{\frac{x}{\sqrt[3]{9b}}}$   $\ans{\displaystyle{\frac{x \sqrt[3]{3b^2}}{3b}}}$

3. Simplify:

1. $\displaystyle{\frac{\displaystyle{\frac{x}{y^2} + \frac{y}{x^2}}}{\displaystyle{\frac{y^2 - yx + x^2}{y}}}}$   $\ans{\displaystyle{\frac{x+y}{x^2 y}}}$

2. $\displaystyle{\frac{24pq^8 r^{-4}}{36p^{-3} q^{-6} r^5}}$   $\ans{\displaystyle{\frac{2p^4 q^{14}}{3r^9}}}$

3. $\displaystyle{\frac{\sqrt[4]{(c+d)^3} \cdot \sqrt{(c+d)}}{\sqrt[5]{(c+d)^4}}}$   $\ans{\displaystyle{\sqrt[20]{(c+d)^9}}}$

4. $\displaystyle{\sqrt[8]{\frac{m^{32}n^{16}}{3^8}}}$   $\ans{\displaystyle{\frac{m^4 n^2}{3}}}$

5. $\displaystyle{\sqrt{y^5} \sqrt[3]{y^2}}$   $\ans{\displaystyle{y^3 \sqrt[6]{y}}}$

6. $\displaystyle{\frac{\displaystyle{1+\frac{1}{x+3}}}{\displaystyle{x-\frac{24}{x-2}}}}$   $\ans{\displaystyle{\frac{x-2}{(x+3)(x-6)}}}$

7. $\displaystyle{\frac{b-a^{-1}}{a-b^{-1}}}$   $\ans{\displaystyle{\frac{b}{a}}}$

8. $\displaystyle{(5xy^4 - 7xy^2 + 4x^2 - 3) - (-3xy^4 + 2xy^2 - 2y + 4)}$   $\ans{\displaystyle{8xy^4 - 9xy^2 + 4x^2 + 2y - 7}}$

9. $\displaystyle{\frac{6c}{c^2-9} - \frac{5c}{c^2-c-6}}$   $\ans{\displaystyle{\frac{c}{(c+3)(c+2)}}}$

10. $\displaystyle{\frac{3x^2 - 12}{x^2 + 4x + 4} \div \frac{x-2}{x+2}}$   $\ans{\displaystyle{3}}$

11. $\displaystyle{\frac{54x^6 y^{-4} z^2}{9x^{-3} y^2 z^{-4}}}$   $\ans{\displaystyle{\frac{6x^9 z^6}{y^6}}}$

12. $\displaystyle{\frac{3}{x-1} + \frac{1}{2x} + \frac{5x}{x^2 - 1}}$   $\ans{\displaystyle{\frac{17x^2 + 6x -1}{2x(x+1)(x-1)}}}$

13. $\displaystyle{(5x^2 - \sqrt{2})^2}$   $\ans{\displaystyle{25x^4 - 10x^2 \sqrt{2} + 2}}$

14. $\displaystyle{8\sqrt{5} + \frac{25}{\sqrt{5}}}$   $\ans{\displaystyle{13\sqrt{5}}}$

15. $\displaystyle{\frac{(2a^{-1} b^2 c)^2}{(a b^{-1})^3}}$   $\ans{\displaystyle{\frac{4b^7 c^2}{a^5}}}$

16. $\displaystyle{\sqrt[3]{a^4} = \frac{\sqrt[3]{a^2}}{\sqrt[3]{a^{-1}}} + \sqrt[3]{\frac{8}{a^{-3}}}}$   $\ans{\displaystyle{a \sqrt[3]{a} + a}}$