Exam 13: Reference: Basic Algebraic Concepts

Simplify each expression. Assume that all variables represent positive real numbers. (a) $(-2)^{-5}$ (b) $\left(125 y^{6}\right)^{2 / 3}$ (c) $\left(\frac{27}{8}\right)^{-2 / 3}$

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Question 1

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(a) $-\frac{1}{32}$
(b) $25 y^{4}$
(c) $\frac{4}{9}$

Simplify each expression. Assume that all variables represent positive real numbers. (a) $\sqrt[3]{64}$ (b) $\sqrt[4]{\frac{1}{2}}$ (c) $\sqrt{27 u^{4} v^{7}}$ (d) $\frac{\sqrt[5]{x^{2} y^{2}} \sqrt[5]{x}}{\sqrt[5]{y^{3}}}$

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Question 2

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(a) 4
(b) $\frac{\sqrt[4]{8}}{2}$
(c) $3 u^{2} v^{3} \sqrt{3 v}$
(d) $\frac{\sqrt[5]{x^{3} y^{4}}}{y}$

Simplify each expression. Leave answers with exponents. Assume that all variables represent nonzero real numbers. (a) $(-9)^{2} \cdot(-9)^{3} \cdot(-9)^{8}$ (b) $\left(2 x^{2} y^{5}\right)^{3}$ (c) $4(x y)^{0}$ (d) $\left(-\frac{2}{3}\right)^{4}$ (e) $\left(-\frac{p}{q^{3}}\right)^{2}$

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Question 3

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(a) $(-9)^{13}$
(b) $2^{3} x^{6} y^{15}$
(c) 4
(d) $\frac{16}{81}$
(e) $\frac{p^{2}}{q^{6}}$

Simplify each expression. Assume that all variables represent positive real numbers. (a) $(-6)^{-3}$ (b) $\left(25 x^{4} y^{2}\right)^{3 / 2}$ (c) $\left(\frac{16}{81}\right)^{-3 / 4}$

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Question 4

Simplify each expression. Leave answers with exponents. Assume that all variables represent nonzero real numbers. (a) $(-6)^{7} \cdot(-6)^{8} \cdot(-6)^{9}$ (b) $\left(5 x^{5} y^{3}\right)^{3}$ (c) $-6(2 y)^{0}$ (d) $\left(-\frac{4}{7}\right)^{4}$ (e) $-\left(\frac{m^{4}}{p^{6}}\right)^{4}$

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Question 5

Simplify each expression. Leave answers with exponents. Assume that all variables represent nonzero real numbers. (a) $(-2)^{5} \cdot(-2)^{3} \cdot(-2)^{7}$ (b) $\left(2 s^{2} t\right)^{4}$ (c) $\left(-4 x y^{2}\right)^{0}$ (d) $\left(-\frac{2}{3}\right)^{3}$ (e) $\left(\frac{w^{2}}{x^{3}}\right)^{2}$

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Question 6

Simplify each expression. Assume that all variables represent positive real numbers. (a) $3^{-5}$ (b) $\left(27 x^{12}\right)^{2 / 3}$ (c) $\left(\frac{9}{16}\right)^{-3 / 2}$

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Question 7

Write each rational expression in lowest terms, and give the domain. (a) $\frac{33 p^{5}}{11 p^{8}}$ (b) $\frac{9+k}{k^{2}-81}$ (c) $\frac{y^{3}+27}{y^{2}-9}$

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Question 8

Simplify. Assume that all variables represent positive real numbers. (a) $\sqrt[3]{16 x}-\sqrt[3]{2 x}+\sqrt[3]{54 x}$ (b) $(\sqrt[3]{16}+3)(\sqrt[3]{4}+4)$ (c) $(x+\sqrt{y})(x-\sqrt{y})$

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Question 9

Perform the indicated operations. (a) $\frac{2 x^{2}-x-3}{20 x^{4}-30 x^{3}} \cdot \frac{5 x^{10}+25 x^{9}-30 x^{8}}{x^{4}+5 x^{2}-6}$ (b) $\frac{2}{x^{2}-2 x-3}+\frac{1}{x^{2}-7 x+12}$ (c) $\frac{u+2 v}{w-2 z}+\frac{u-2 v}{2 z-w}$ (d) $\frac{r-\frac{2}{r}}{r-\frac{1}{r}}$

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Question 10

Perform the indicated operations. Write your answers with only positive exponents. Assume that all variables represent positive real numbers. (a) $\frac{6^{-5}}{6^{-1} \cdot 6^{-2}}$ (b) $(y+3)^{-1 / 6}(y+3)^{-5 / 12}$ (c) $\left(r^{-1} s^{2 / 3}\right)^{6}\left(8 r^{6} s^{-2}\right)^{-1 / 3}$

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Question 11

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

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Question 12

Perform the indicated operations. (a) $\left(x^{2}-5 x+2\right)+\left(2 x-3 x^{2}\right)-2 x(2 x+1)$ (b) $(6-5 x)^{2}$ (c) $(r+3)\left(2 r^{2}+9 r-5\right)$ (d) $(2 t-5)(5+2 t)$ (e) $[3 y+(4-2 y)]^{2}$

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Question 13

Simplify each expression. Assume that all variables represent positive real numbers. (a) $\sqrt[5]{32}$ (b) $\sqrt[3]{\frac{3}{2}}$ (c) $\sqrt{75 x^{8} y^{3}}$ (d) $\frac{\sqrt[3]{m n} \sqrt[3]{m}}{\sqrt[3]{n^{2}}}$

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Question 14

Perform the indicated operations. (a) $\frac{3 x^{3}-5 x^{2}-2 x}{30 x^{4}+10 x^{3}} \cdot \frac{4 x^{5}+32 x^{4}+48 x^{3}}{x^{4}-11 x^{2}+28}$ (b) $\frac{x}{x^{2}-5 x+6}-\frac{2 x}{2 x^{2}-3 x-2}$ (c) $\frac{3 a-2 b}{2 x-y}-\frac{2 b-3 a}{y-2 x}$ (d) $\frac{t+\frac{3}{t}}{t-\frac{9}{t}}$

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Question 15

Rationalize the denominator of $\frac{21}{\sqrt{13}-\sqrt{6}}$ and simplify.

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Question 16

Perform the indicated operations. (a) $\left(3 x^{2}-4 x\right)-\left(2+x-x^{2}\right)+(x+1)(2 x+1)$ (b) $(3 w-5)^{2}$ (c) $(3 t+2)\left(t^{2}-t+5\right)$ (d) $(3-2 x)(3+2 x)$ (e) $[(p-2)+2 p]^{2}$

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Question 17

Simplify. Assume that all variables represent positive real numbers. (a) $\sqrt{75 x}+\sqrt{3 x}-\sqrt{12 x}$ (b) $(\sqrt[3]{3}+2)(\sqrt[3]{9}+4)$ (c) $(\sqrt{u}-\sqrt{v})(\sqrt{u}+\sqrt{v})$

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Question 18

Perform the indicated operations. (a) $\left(3 y^{2}-5 y+7\right)-\left(2+3 y^{2}\right)+2 y(3 y-5)$ (b) $(5 r+2)^{2}$ (c) $(2 u-3)\left(4 u^{2}+6 u+9\right)$ (d) $(x+2 y)(x-2 y)$ (e) $[1+(3 t-5)]^{2}$

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Question 19

Simplify each expression. Leave answers with exponents. Assume that all variables represent nonzero real numbers. (a) $3^{3} \cdot 3^{2} \cdot(-3)^{8}$ (b) $\left(5 p^{3} q^{2}\right)^{4}$ (c) $-(4 t)^{0}$ (d) $\left(-\frac{1}{6}\right)^{5}$ (e) $\left(\frac{x^{3}}{y^{4}}\right)^{5}$

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Question 20

Simplify each expression. Assume that all variables represent positive real numbers. (a) $(-2)^{-5}$ (b) $\left(125 y^{6}\right)^{2 / 3}$ (c) $\left(\frac{27}{8}\right)^{-2 / 3}$

Simplify each expression. Assume that all variables represent positive real numbers. (a) $\sqrt[3]{64}$ (b) $\sqrt[4]{\frac{1}{2}}$ (c) $\sqrt{27 u^{4} v^{7}}$ (d) $\frac{\sqrt[5]{x^{2} y^{2}} \sqrt[5]{x}}{\sqrt[5]{y^{3}}}$

Simplify each expression. Leave answers with exponents. Assume that all variables represent nonzero real numbers. (a) $(-9)^{2} \cdot(-9)^{3} \cdot(-9)^{8}$ (b) $\left(2 x^{2} y^{5}\right)^{3}$ (c) $4(x y)^{0}$ (d) $\left(-\frac{2}{3}\right)^{4}$ (e) $\left(-\frac{p}{q^{3}}\right)^{2}$

Simplify each expression. Assume that all variables represent positive real numbers. (a) $(-6)^{-3}$ (b) $\left(25 x^{4} y^{2}\right)^{3 / 2}$ (c) $\left(\frac{16}{81}\right)^{-3 / 4}$

Simplify each expression. Leave answers with exponents. Assume that all variables represent nonzero real numbers. (a) $(-6)^{7} \cdot(-6)^{8} \cdot(-6)^{9}$ (b) $\left(5 x^{5} y^{3}\right)^{3}$ (c) $-6(2 y)^{0}$ (d) $\left(-\frac{4}{7}\right)^{4}$ (e) $-\left(\frac{m^{4}}{p^{6}}\right)^{4}$

Simplify each expression. Assume that all variables represent positive real numbers. (a) $3^{-5}$ (b) $\left(27 x^{12}\right)^{2 / 3}$ (c) $\left(\frac{9}{16}\right)^{-3 / 2}$

Write each rational expression in lowest terms, and give the domain. (a) $\frac{33 p^{5}}{11 p^{8}}$ (b) $\frac{9+k}{k^{2}-81}$ (c) $\frac{y^{3}+27}{y^{2}-9}$

Simplify. Assume that all variables represent positive real numbers. (a) $\sqrt[3]{16 x}-\sqrt[3]{2 x}+\sqrt[3]{54 x}$ (b) $(\sqrt[3]{16}+3)(\sqrt[3]{4}+4)$ (c) $(x+\sqrt{y})(x-\sqrt{y})$

Perform the indicated operations. (a) $\frac{2 x^{2}-x-3}{20 x^{4}-30 x^{3}} \cdot \frac{5 x^{10}+25 x^{9}-30 x^{8}}{x^{4}+5 x^{2}-6}$ (b) $\frac{2}{x^{2}-2 x-3}+\frac{1}{x^{2}-7 x+12}$ (c) $\frac{u+2 v}{w-2 z}+\frac{u-2 v}{2 z-w}$ (d) $\frac{r-\frac{2}{r}}{r-\frac{1}{r}}$

Perform the indicated operations. Write your answers with only positive exponents. Assume that all variables represent positive real numbers. (a) $\frac{6^{-5}}{6^{-1} \cdot 6^{-2}}$ (b) $(y+3)^{-1 / 6}(y+3)^{-5 / 12}$ (c) $\left(r^{-1} s^{2 / 3}\right)^{6}\left(8 r^{6} s^{-2}\right)^{-1 / 3}$

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

Perform the indicated operations. (a) $\left(x^{2}-5 x+2\right)+\left(2 x-3 x^{2}\right)-2 x(2 x+1)$ (b) $(6-5 x)^{2}$ (c) $(r+3)\left(2 r^{2}+9 r-5\right)$ (d) $(2 t-5)(5+2 t)$ (e) $[3 y+(4-2 y)]^{2}$

Simplify each expression. Assume that all variables represent positive real numbers. (a) $\sqrt[5]{32}$ (b) $\sqrt[3]{\frac{3}{2}}$ (c) $\sqrt{75 x^{8} y^{3}}$ (d) $\frac{\sqrt[3]{m n} \sqrt[3]{m}}{\sqrt[3]{n^{2}}}$

Perform the indicated operations. (a) $\frac{3 x^{3}-5 x^{2}-2 x}{30 x^{4}+10 x^{3}} \cdot \frac{4 x^{5}+32 x^{4}+48 x^{3}}{x^{4}-11 x^{2}+28}$ (b) $\frac{x}{x^{2}-5 x+6}-\frac{2 x}{2 x^{2}-3 x-2}$ (c) $\frac{3 a-2 b}{2 x-y}-\frac{2 b-3 a}{y-2 x}$ (d) $\frac{t+\frac{3}{t}}{t-\frac{9}{t}}$

Rationalize the denominator of $\frac{21}{\sqrt{13}-\sqrt{6}}$ and simplify.

Perform the indicated operations. (a) $\left(3 x^{2}-4 x\right)-\left(2+x-x^{2}\right)+(x+1)(2 x+1)$ (b) $(3 w-5)^{2}$ (c) $(3 t+2)\left(t^{2}-t+5\right)$ (d) $(3-2 x)(3+2 x)$ (e) $[(p-2)+2 p]^{2}$

Simplify. Assume that all variables represent positive real numbers. (a) $\sqrt{75 x}+\sqrt{3 x}-\sqrt{12 x}$ (b) $(\sqrt[3]{3}+2)(\sqrt[3]{9}+4)$ (c) $(\sqrt{u}-\sqrt{v})(\sqrt{u}+\sqrt{v})$

Perform the indicated operations. (a) $\left(3 y^{2}-5 y+7\right)-\left(2+3 y^{2}\right)+2 y(3 y-5)$ (b) $(5 r+2)^{2}$ (c) $(2 u-3)\left(4 u^{2}+6 u+9\right)$ (d) $(x+2 y)(x-2 y)$ (e) $[1+(3 t-5)]^{2}$

Simplify each expression. Leave answers with exponents. Assume that all variables represent nonzero real numbers. (a) $3^{3} \cdot 3^{2} \cdot(-3)^{8}$ (b) $\left(5 p^{3} q^{2}\right)^{4}$ (c) $-(4 t)^{0}$ (d) $\left(-\frac{1}{6}\right)^{5}$ (e) $\left(\frac{x^{3}}{y^{4}}\right)^{5}$

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Exam 13: Reference: Basic Algebraic Concepts

Simplify each expression. Assume that all variables represent positive real numbers. (a) (−2)−5(-2)^{-5}(−2)−5 (b) (125y6)2/3\left(125 y^{6}\right)^{2 / 3}(125y6)2/3 (c) (278)−2/3\left(\frac{27}{8}\right)^{-2 / 3}(827​)−2/3

Simplify each expression. Assume that all variables represent positive real numbers. (a) 643\sqrt[3]{64}364​ (b) 124\sqrt[4]{\frac{1}{2}}421​​ (c) 27u4v7\sqrt{27 u^{4} v^{7}}27u4v7​ (d) x2y25x5y35\frac{\sqrt[5]{x^{2} y^{2}} \sqrt[5]{x}}{\sqrt[5]{y^{3}}}5y3​5x2y2​5x​​

Simplify each expression. Assume that all variables represent positive real numbers. (a) (−6)−3(-6)^{-3}(−6)−3 (b) (25x4y2)3/2\left(25 x^{4} y^{2}\right)^{3 / 2}(25x4y2)3/2 (c) (1681)−3/4\left(\frac{16}{81}\right)^{-3 / 4}(8116​)−3/4

Simplify each expression. Assume that all variables represent positive real numbers. (a) 3−53^{-5}3−5 (b) (27x12)2/3\left(27 x^{12}\right)^{2 / 3}(27x12)2/3 (c) (916)−3/2\left(\frac{9}{16}\right)^{-3 / 2}(169​)−3/2

Write each rational expression in lowest terms, and give the domain. (a) 33p511p8\frac{33 p^{5}}{11 p^{8}}11p833p5​ (b) 9+kk2−81\frac{9+k}{k^{2}-81}k2−819+k​ (c) y3+27y2−9\frac{y^{3}+27}{y^{2}-9}y2−9y3+27​

Simplify. Assume that all variables represent positive real numbers. (a) 16x3−2x3+54x3\sqrt[3]{16 x}-\sqrt[3]{2 x}+\sqrt[3]{54 x}316x​−32x​+354x​ (b) (163+3)(43+4)(\sqrt[3]{16}+3)(\sqrt[3]{4}+4)(316​+3)(34​+4) (c) (x+y)(x−y)(x+\sqrt{y})(x-\sqrt{y})(x+y​)(x−y​)

Simplify each expression. Assume that all variables represent positive real numbers. (a) 325\sqrt[5]{32}532​ (b) 323\sqrt[3]{\frac{3}{2}}323​​ (c) 75x8y3\sqrt{75 x^{8} y^{3}}75x8y3​ (d) mn3m3n23\frac{\sqrt[3]{m n} \sqrt[3]{m}}{\sqrt[3]{n^{2}}}3n2​3mn​3m​​

Rationalize the denominator of 2113−6\frac{21}{\sqrt{13}-\sqrt{6}}13​−6​21​ and simplify.

Simplify. Assume that all variables represent positive real numbers. (a) 75x+3x−12x\sqrt{75 x}+\sqrt{3 x}-\sqrt{12 x}75x​+3x​−12x​ (b) (33+2)(93+4)(\sqrt[3]{3}+2)(\sqrt[3]{9}+4)(33​+2)(39​+4) (c) (u−v)(u+v)(\sqrt{u}-\sqrt{v})(\sqrt{u}+\sqrt{v})(u​−v​)(u​+v​)

Linear Functions, Equations, and Inequalities

Analysis of Graphs of Functions

Polynomial Functions

Rational, Power, and Root Functions

Inverse, Exponential, and Logarithmic Functions

Systems and Matrices

Analytic Geometry and Nonlinear Systems

The Unit Circle and Functions of Trigonometry

Trigonometric Identities and Equations

Applications of Trigonometry and Vectors

Further Topics in Algebra

Limits, Derivatives, and Definite Integrals

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Simplify each expression. Assume that all variables represent positive real numbers. (a) $(-2)^{-5}$ (b) $\left(125 y^{6}\right)^{2 / 3}$ (c) $\left(\frac{27}{8}\right)^{-2 / 3}$

Simplify each expression. Assume that all variables represent positive real numbers. (a) $\sqrt[3]{64}$ (b) $\sqrt[4]{\frac{1}{2}}$ (c) $\sqrt{27 u^{4} v^{7}}$ (d) $\frac{\sqrt[5]{x^{2} y^{2}} \sqrt[5]{x}}{\sqrt[5]{y^{3}}}$

Simplify each expression. Assume that all variables represent positive real numbers. (a) $(-6)^{-3}$ (b) $\left(25 x^{4} y^{2}\right)^{3 / 2}$ (c) $\left(\frac{16}{81}\right)^{-3 / 4}$

Simplify each expression. Assume that all variables represent positive real numbers. (a) $3^{-5}$ (b) $\left(27 x^{12}\right)^{2 / 3}$ (c) $\left(\frac{9}{16}\right)^{-3 / 2}$

Write each rational expression in lowest terms, and give the domain. (a) $\frac{33 p^{5}}{11 p^{8}}$ (b) $\frac{9+k}{k^{2}-81}$ (c) $\frac{y^{3}+27}{y^{2}-9}$

Simplify. Assume that all variables represent positive real numbers. (a) $\sqrt[3]{16 x}-\sqrt[3]{2 x}+\sqrt[3]{54 x}$ (b) $(\sqrt[3]{16}+3)(\sqrt[3]{4}+4)$ (c) $(x+\sqrt{y})(x-\sqrt{y})$

Simplify each expression. Assume that all variables represent positive real numbers. (a) $\sqrt[5]{32}$ (b) $\sqrt[3]{\frac{3}{2}}$ (c) $\sqrt{75 x^{8} y^{3}}$ (d) $\frac{\sqrt[3]{m n} \sqrt[3]{m}}{\sqrt[3]{n^{2}}}$

Rationalize the denominator of $\frac{21}{\sqrt{13}-\sqrt{6}}$ and simplify.

Simplify. Assume that all variables represent positive real numbers. (a) $\sqrt{75 x}+\sqrt{3 x}-\sqrt{12 x}$ (b) $(\sqrt[3]{3}+2)(\sqrt[3]{9}+4)$ (c) $(\sqrt{u}-\sqrt{v})(\sqrt{u}+\sqrt{v})$