Exam 13: Reference: Basic Algebraic Concepts

Factor completely. (a) $3 x^{2}-4 x-15$ (b) $x^{3}-64$ (c) $x^{2}+w x-20 w^{2}$ (d) $x^{3} y^{2}+4 x^{3}+8 y^{2}+32$

Simplify each expression. Assume that all variables represent positive real numbers. (a) $\sqrt[5]{243}$ (b) $\sqrt[3]{\frac{5}{9}}$ (c) $\sqrt{28 x^{6} y^{3}}$ (d) $\frac{\sqrt[5]{p q} \sqrt[5]{p q^{2}}}{\sqrt[5]{p^{4}}}$

Perform the indicated operations. (a) $\left(2 x^{2}+6 x-5\right)-\left(2 x-3 x^{2}\right)+x(3 x+2)$ (b) $(2 u+7)^{2}$ (c) $(y-2)\left(2 y^{2}-3 y+5\right)$ (d) $(6 t-7)(6 t+7)$ (e) $[(2 p-3)+6]^{2}$

Perform the indicated operations. Write your answers with only positive exponents. Assume that all variables represent positive real numbers. (a) $\frac{3^{-5} \cdot 3^{-6}}{3^{-7}}$ (b) $(2 x+1)^{-1 / 3}(2 x+1)^{-5 / 6}$ (c) $\left(x^{-2} y^{1 / 3}\right)^{6}\left(16 x^{5} y^{-4}\right)^{-1 / 2}$

Perform the indicated operations. Write your answers with only positive exponents. Assume that all variables represent positive real numbers. (a) $\frac{5^{-3} \cdot 5^{-5}}{5^{-2} \cdot 5^{-1}}$ (b) $(x+4)^{2 / 5}(x+4)^{-9 / 10}$ (c) $\left(2 u^{2} v^{-1 / 2}\right)^{4}\left(4 u^{10} v^{3}\right)^{-1 / 2}$

Perform the indicated operations. (a) $\frac{2 x^{2}-2 x-4}{24 x^{4}+8 x^{3}} \cdot \frac{6 x^{8}+8 x^{7}+2 x^{6}}{x^{4}-5 x^{2}+4}$ (b) $\frac{2 x}{x^{2}+3 x+2}+\frac{x}{x^{2}-1}$ (c) $\frac{m+2 n}{x-y}+\frac{m-2 n}{y-x}$ (d) $\frac{p+\frac{1}{p}}{p-\frac{1}{p}}$

Simplify each expression. Assume that all variables represent positive real numbers. (a) $\sqrt[4]{625}$ (b) $\sqrt[3]{\frac{3}{5}}$ (c) $\sqrt{27 x^{5} y^{6}}$ (d) $\frac{\sqrt[4]{p^{2} q} \sqrt[4]{q}}{\sqrt[4]{p^{3}}}$

Factor completely. (a) $10 s^{2}+29 s-21$ (b) $z^{3}-1$ (c) $2 x^{2}-x y-6 y^{2}$ (d) $x^{2} y^{2}-9 x^{2}-4 y^{2}+36$

Write each rational expression in lowest terms, and give the domain. (a) $\frac{14 y^{2}}{10 y^{5}}$ (b) $\frac{2-m}{m^{2}-4}$ (c) $\frac{t^{2}-9}{3 t^{2}+10 t+3}$

Simplify. Assume that all variables represent positive real numbers. (a) $\sqrt{9 x}+\sqrt{36 x}-\sqrt{49 x}$ (b) $(2-\sqrt[3]{2})(2+\sqrt[3]{4})$ (c) $(\sqrt{x y}-1)(\sqrt{x y}+1)$

Simplify each expression. Assume that all variables represent positive real numbers. (a) $(-3)^{-3}$ (b) $\left(8 x^{3}\right)^{2 / 3}$ (c) $\left(\frac{27}{125}\right)^{-1 / 3}$

Write each rational expression in lowest terms, and give the domain. (a) $\frac{12 y^{5}}{4 y^{13}}$ (b) $\frac{2-t}{4-t^{2}}$ (c) $\frac{x^{2}+5 x+6}{x^{2}-2 x-15}$

Perform the indicated operations. (a) $\frac{2 x^{2}+9 x-5}{18 x^{6}-9 x^{5}} \cdot \frac{3 x^{6}-9 x^{5}-30 x^{4}}{x^{4}-26 x^{2}+25}$ (b) $\frac{2 x}{x^{2}-x-12}-\frac{x}{2 x^{2}-7 x-4}$ (c) $\frac{2 a+b}{3 u-v}-\frac{2 a-b}{v-3 u}$ (d) $\frac{\frac{3}{k}-k}{k-\frac{9}{k}}$

Factor completely. (a) $6 t^{2}-7 t-3$ (b) $x^{4}-625$ (c) $z^{2}-5 z k-14 k^{2}$ (d) $x^{3} y^{2}-x^{3}+y^{2}-1$

Write each rational expression in lowest terms, and give the domain. (a) $\frac{3 x^{7}}{9 x^{9}}$ (b) $\frac{3 x+1}{9 x^{2}-1}$ (c) $\frac{x^{2}-2 x-3}{2 x^{2}-x-3}$

Simplify. Assume that all variables represent positive real numbers.. (a) $\sqrt{20 x}-\sqrt{5 x}+\sqrt{45 x}$ (b) $(\sqrt[5]{4}+2)(\sqrt[5]{8}+3)$ (c) $(\sqrt{5 x}+\sqrt{3 y})(\sqrt{5 x}-\sqrt{3 y})$

Factor completely. (a) $6 x^{2}+19 x+10$ (b) $x^{3}+27$ (c) $p^{2}-p q-6 q^{2}$ (d) $x^{2} y^{3}-8 x^{2}-4 y^{3}+32$

Rationalize the denominator of $\frac{3}{2 \sqrt{10}-\sqrt{7}}$ and simplify.

Rationalize the denominator of $\frac{1}{3 \sqrt{2}+2 \sqrt{3}}$ and simplify.

Rationalize the denominator of $\frac{6}{\sqrt{15}+\sqrt{6}}$ and simplify.