Deck 13: Reference: Basic Algebraic Concepts

Question 1

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) $\left(-\frac{p}{q^{3}}\right)^{2}$

Answer 1

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

Question 2

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) $[1+(3 t-5)]^{2}$

Answer 2

(a) $6 y^{2}-15 y+5$
(b) $25 r^{2}+20 r+4$
(c) $8 u^{3}-27$
(d) $x^{2}-4 y^{2}$
(e) $9 t^{2}-24 t+16$

Question 3

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

Answer 3

(a) $(2 x+5)(3 x+2)$
(b) $(x+3)\left(x^{2}-3 x+9\right)$
(c) $(p+2 q)(p-3 q)$
(d) $(x-2)(x+2)(y-2)\left(y^{2}+2 y+4\right)$

Question 4

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) $\frac{x^{2}+5 x+6}{x^{2}-2 x-15}$

Question 5

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) $\frac{\frac{3}{k}-k}{k-\frac{9}{k}}$

Question 6

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

Question 7

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) $\left(m^{-2} n^{1 / 2}\right)^{3}\left(9 m n^{-3}\right)^{-1 / 2}$

Question 8

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) $\frac{\sqrt[3]{m n} \sqrt[3]{m}}{\sqrt[3]{n^{2}}}$

Question 9

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) $(\sqrt{x y}-1)(\sqrt{x y}+1)$

Question 10

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

Question 11

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) $\left(\frac{w^{2}}{x^{3}}\right)^{2}$

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) $[3 y+(4-2 y)]^{2}$

Question 13

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

Question 14

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) $\frac{x^{2}-2 x-3}{2 x^{2}-x-3}$

Question 15

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) $\frac{p+\frac{1}{p}}{p-\frac{1}{p}}$

Question 16

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

Question 17

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) $\left(r^{-1} s^{2 / 3}\right)^{6}\left(8 r^{6} s^{-2}\right)^{-1 / 3}$

Question 18

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) $\frac{\sqrt[4]{p^{2} q} \sqrt[4]{q}}{\sqrt[4]{p^{3}}}$

Question 19

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) $(\sqrt{u}-\sqrt{v})(\sqrt{u}+\sqrt{v})$

Question 20

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

Question 21

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) $\left(\frac{x^{3}}{y^{4}}\right)^{5}$

Question 22

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) $[(2 p-3)+6]^{2}$

Question 23

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

Question 24

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) $\frac{t^{2}-9}{3 t^{2}+10 t+3}$

Question 25

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) $\frac{t+\frac{3}{t}}{t-\frac{9}{t}}$

Question 26

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

Question 27

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) $\left(2 u^{2} v^{-1 / 2}\right)^{4}\left(4 u^{10} v^{3}\right)^{-1 / 2}$

Question 28

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

Question 29

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) $(x+\sqrt{y})(x-\sqrt{y})$

Question 30

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

Question 31

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) $-\left(\frac{m^{4}}{p^{6}}\right)^{4}$

Question 32

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) $[(p-2)+2 p]^{2}$

Question 33

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

Question 34

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) $\frac{y^{3}+27}{y^{2}-9}$

Question 35

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) $\frac{r-\frac{2}{r}}{r-\frac{1}{r}}$

Question 36

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) $\left(\frac{16}{81}\right)^{-3 / 4}$

Question 37

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) $\left(x^{-2} y^{1 / 3}\right)^{6}\left(16 x^{5} y^{-4}\right)^{-1 / 2}$

Question 38

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) $\frac{\sqrt[5]{p q} \sqrt[5]{p q^{2}}}{\sqrt[5]{p^{4}}}$

Question 39

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) $(\sqrt{5 x}+\sqrt{3 y})(\sqrt{5 x}-\sqrt{3 y})$

Question 40

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