Question 1

Multiply, then simplify the product. Assume that all variables represent positive real numbers.
-$(\sqrt{5 x}+5)(\sqrt{3 x}+3)$

Accepted Answer

A)  $x \sqrt{15}+\sqrt{3 x}+15$ 
B)  $x \sqrt{15}+15$ 
C)  $x \sqrt{3}+15$ 
D)  $x \sqrt{15}+3 \sqrt{5 x}+5 \sqrt{3 x}+15$ 
A)  $x \sqrt{15}+\sqrt{3 x}+15$ 
B)  $x \sqrt{15}+15$ 
C)  $x \sqrt{3}+15$ 
D)  $x \sqrt{15}+3 \sqrt{5 x}+5 \sqrt{3 x}+15$

Question 2

Simplify. Assume that all variables represent positive real numbers.
-$\sqrt{500 k^{7} q^{8}}$

Accepted Answer

A)  $10 k^{3} q^{4} \sqrt{5 k}$ 
B)  $10 k^{2} q^{4} \sqrt{5 k}$ 
C)  $10 k^{7} q^{8} \sqrt{5 k}$ 
D)  $10 \mathrm{k}^{3} \mathrm{q}^{4} \sqrt{5}$ 
A)  $10 k^{3} q^{4} \sqrt{5 k}$ 
B)  $10 k^{2} q^{4} \sqrt{5 k}$ 
C)  $10 k^{7} q^{8} \sqrt{5 k}$ 
D)  $10 \mathrm{k}^{3} \mathrm{q}^{4} \sqrt{5}$

Question 3

Choose the one alternative that best completes the statement or answers the question.
Solve the equation
-$\sqrt{7 \mathrm{x}-2}-9=0$

Accepted Answer

A)  $\left\{\frac{83}{7}\right\}$ 
B)  $\left\{\frac{11}{7}\right\}$ 
C)  $\{81\}$ 
D)  $\varnothing$ 
A)  $\left\{\frac{83}{7}\right\}$ 
B)  $\left\{\frac{11}{7}\right\}$ 
C)  $\{81\}$ 
D)  $\varnothing$

Question 4

$(9-4 \mathrm{i})^{2}$

Accepted Answer

A)  $97-72 \mathrm{i}$ 
B)  $65-72 \mathrm{i}$ 
C)  $81-88 \mathrm{i}$ 
D)  81 - 56i 
A)  $97-72 \mathrm{i}$ 
B)  $65-72 \mathrm{i}$ 
C)  $81-88 \mathrm{i}$ 
D)  81 - 56i

Question 5

Use a calculator to approximate the root to the nearest thousandth.
-$\sqrt{251}$

Accepted Answer

A)  15.858 
B)  15 
C)  15.843 
D)  15.830 
A)  15.858 
B)  15 
C)  15.843 
D)  15.830

Question 6

Find the decimal approximation for the radical. Round the answer to three decimal places.
-$\sqrt[3]{67}$

Accepted Answer

A)  66.996 
B)  4.066 
C)  67.009 
D)  4.062 
A)  66.996 
B)  4.066 
C)  67.009 
D)  4.062

Question 7

Choose the one alternative that best completes the statement or answers the question.
Solve the equation
-$\sqrt[4]{\mathrm{x}-4}+10=0$

Accepted Answer

A)  $\left\{-\frac{5}{2}\right\}$ 
B)  $\{2500\}$ 
C)  $\{-2500\}$ 
D)  $\varnothing$ 
A)  $\left\{-\frac{5}{2}\right\}$ 
B)  $\{2500\}$ 
C)  $\{-2500\}$ 
D)  $\varnothing$

Question 8

Write the expression in the form a + bi.
-$\frac{6+2 i}{9-3 i}$

Accepted Answer

A)  $\frac{5}{6}-\frac{1}{12} \mathrm{i}$ 
B)  $\frac{14}{15}$ 
C)  $-\frac{48}{65}-\frac{36}{65} \mathrm{i}$ 
D)  $\frac{8}{15}+\frac{2}{5} \mathrm{i}$ 
A)  $\frac{5}{6}-\frac{1}{12} \mathrm{i}$ 
B)  $\frac{14}{15}$ 
C)  $-\frac{48}{65}-\frac{36}{65} \mathrm{i}$ 
D)  $\frac{8}{15}+\frac{2}{5} \mathrm{i}$

Question 9

Simplify. Assume that all variables represent positive real numbers.
-$\sqrt[4]{\frac{16}{\mathrm{y}}}$

Accepted Answer

A)  $\frac{2 \sqrt[4]{y^{3}}}{y^{2}}$ 
B)  $\frac{2 \sqrt[4]{y^{3}}}{y}$ 
C)  $\frac{2 \sqrt[4]{\mathrm{y}}}{\mathrm{y}}$ 
D)  $\frac{2 \sqrt[4]{y^{2}}}{y}$ 
A)  $\frac{2 \sqrt[4]{y^{3}}}{y^{2}}$ 
B)  $\frac{2 \sqrt[4]{y^{3}}}{y}$ 
C)  $\frac{2 \sqrt[4]{\mathrm{y}}}{\mathrm{y}}$ 
D)  $\frac{2 \sqrt[4]{y^{2}}}{y}$

Question 10

Simplify the radical. Assume that all variables represent positive real numbers
-$\sqrt[4]{\frac{6 y^{7}}{7776 y^{3}}}$

Accepted Answer

A)  $\frac{y}{6}$ 
B)  $6 y$ 
C)  $\frac{y^{2}}{36}$ 
D)  $\frac{\mathrm{y}}{\sqrt[4]{216}}$ 
A)  $\frac{y}{6}$ 
B)  $6 y$ 
C)  $\frac{y^{2}}{36}$ 
D)  $\frac{\mathrm{y}}{\sqrt[4]{216}}$

Question 11

Find the distance between the pair of points.
-$(2 \sqrt{2}, 2 \sqrt{11})$ and $(9 \sqrt{2},-2 \sqrt{11})$

Accepted Answer

A)  274 
B)  $2 \sqrt{55}$ 
C)  220 
D)  $\sqrt{274}$ 
A)  274 
B)  $2 \sqrt{55}$ 
C)  220 
D)  $\sqrt{274}$

Question 12

Simplify the expression involving rational exponents.
-$(-8)^{2 / 3}$

Accepted Answer

A)  -2 
B)  -4 
C)  4 
D)  Not a real number 
A)  -2 
B)  -4 
C)  4 
D)  Not a real number

Question 13

Choose the one alternative that best completes the statement or answers the question.
Find the root if it is a real number.
-$\sqrt[5]{-243}$

Accepted Answer

A)  3 
B)  -3 
C)  15 
D)  -15 
A)  3 
B)  -3 
C)  15 
D)  -15

Question 14

Solve the formula for the indicated variable.
-$T=2 \pi \sqrt{\frac{L}{K}}$ for $L$

Accepted Answer

A)  $\mathrm{L}=\frac{\mathrm{KT}^{2}}{2 \pi}$ 
B)  $\mathrm{L}=\frac{\mathrm{K}^{2} \mathrm{~T}^{2}}{4 \pi^{2}}$ 
C)  $\mathrm{L}=\frac{\mathrm{KT}}{2 \pi}$ 
D)  $\mathrm{L}=\frac{\mathrm{KT}^{2}}{4 \pi^{2}}$ 
A)  $\mathrm{L}=\frac{\mathrm{KT}^{2}}{2 \pi}$ 
B)  $\mathrm{L}=\frac{\mathrm{K}^{2} \mathrm{~T}^{2}}{4 \pi^{2}}$ 
C)  $\mathrm{L}=\frac{\mathrm{KT}}{2 \pi}$ 
D)  $\mathrm{L}=\frac{\mathrm{KT}^{2}}{4 \pi^{2}}$

Question 15

The following table gives data on two different solar modules being developed for use in roofing.
 
You must determine the size of the frame needed to support each panel on a roof. (Note: The sides of each frame will form a right triangle, and the hypotenuse of the triangle will be the width of the panel.) Use the Pythagorean theorem to find the dimensions of the legs for each frame if the legs have equal length. Round your answers to the nearest tenth when necessary.

Accepted Answer

A)  ABC-01: 3 in.;
$\mathrm{ABC}-02: 3.3 \mathrm{in}$. 
B)  ABC-01: 12.7 in.;
$\mathrm{ABC}-02: 15.6$ in. 
C)  ABC-01: 9 in.;
$\mathrm{ABC}-02: 11$ in. 
D)  ABC-01: 4.5 in.;
$\mathrm{ABC}-02: 5.5 \mathrm{in}$. 
A)  ABC-01: 3 in.;
$\mathrm{ABC}-02: 3.3 \mathrm{in}$. 
B)  ABC-01: 12.7 in.;
$\mathrm{ABC}-02: 15.6$ in. 
C)  ABC-01: 9 in.;
$\mathrm{ABC}-02: 11$ in. 
D)  ABC-01: 4.5 in.;
$\mathrm{ABC}-02: 5.5 \mathrm{in}$.

Question 16

Simplify by first converting to rational exponents. Assume that all variables represent positive real numbers.
-$\sqrt[3]{5^{12}}$

Accepted Answer

A)  $\sqrt[3]{5^{12}}$ 
B)  625 
C)  125 
D)  3125 
A)  $\sqrt[3]{5^{12}}$ 
B)  625 
C)  125 
D)  3125

Question 17

Choose the one alternative that best completes the statement or answers the question.
Solve the equation
-$\sqrt{10+x}+\sqrt{11-5 x}=-1$

Accepted Answer

A)  $\varnothing$ 
B)  $\{1\}$ 
C)  $\left\{\frac{11}{2}\right\}$ 
D)  $\{0\}$ 
A)  $\varnothing$ 
B)  $\{1\}$ 
C)  $\left\{\frac{11}{2}\right\}$ 
D)  $\{0\}$

Question 18

Choose the one alternative that best completes the statement or answers the question.
Simplify. Assume that all variables represent positive real numbers.
-$-17 \sqrt{14}-10 \sqrt{14}$

Accepted Answer

A)  $26 \sqrt{14}$ 
B)  $-27 \sqrt{14}$ 
C)  $-7 \sqrt{14}$ 
D)  $8 \sqrt{14}$ 
A)  $26 \sqrt{14}$ 
B)  $-27 \sqrt{14}$ 
C)  $-7 \sqrt{14}$ 
D)  $8 \sqrt{14}$

Question 19

Choose the one alternative that best completes the statement or answers the question.
Find the root if it is a real number.
-$\sqrt[4]{-625}$

Accepted Answer

A)  -5 
B)  Not a real number 
C)  5 
D)  -25 
A)  -5 
B)  Not a real number 
C)  5 
D)  -25

Question 20

Rewrite the expressions with rational exponents as radical expressions, and then solve the equation.
-$(10+4 t)^{1 / 3}-(7+6 t)^{1 / 3}=0$

Accepted Answer

A)  $\left\{-\frac{3}{2}\right\}$ 
B)  $\left\{\frac{3}{2}\right\}$ 
C)  $\varnothing$ 
D)  $\left\{\frac{2}{3}\right\}$ 
A)  $\left\{-\frac{3}{2}\right\}$ 
B)  $\left\{\frac{3}{2}\right\}$ 
C)  $\varnothing$ 
D)  $\left\{\frac{2}{3}\right\}$

Multiply, then simplify the product. Assume that all variables represent positive real numbers. - $(\sqrt{5 x}+5)(\sqrt{3 x}+3)$

Simplify. Assume that all variables represent positive real numbers. - $\sqrt{500 k^{7} q^{8}}$

Choose the one alternative that best completes the statement or answers the question. Solve the equation - $\sqrt{7 \mathrm{x}-2}-9=0$

$(9-4 \mathrm{i})^{2}$

Use a calculator to approximate the root to the nearest thousandth. - $\sqrt{251}$

Find the decimal approximation for the radical. Round the answer to three decimal places. - $\sqrt[3]{67}$

Choose the one alternative that best completes the statement or answers the question. Solve the equation - $\sqrt[4]{\mathrm{x}-4}+10=0$

Write the expression in the form a + bi. - $\frac{6+2 i}{9-3 i}$

Simplify. Assume that all variables represent positive real numbers. - $\sqrt[4]{\frac{16}{\mathrm{y}}}$

Simplify the radical. Assume that all variables represent positive real numbers - $\sqrt[4]{\frac{6 y^{7}}{7776 y^{3}}}$

Find the distance between the pair of points. - $(2 \sqrt{2}, 2 \sqrt{11})$ and $(9 \sqrt{2},-2 \sqrt{11})$

Simplify the expression involving rational exponents. - $(-8)^{2 / 3}$

Choose the one alternative that best completes the statement or answers the question. Find the root if it is a real number. - $\sqrt[5]{-243}$

Solve the formula for the indicated variable. - $T=2 \pi \sqrt{\frac{L}{K}}$ for $L$

Simplify by first converting to rational exponents. Assume that all variables represent positive real numbers. - $\sqrt[3]{5^{12}}$

Choose the one alternative that best completes the statement or answers the question. Solve the equation - $\sqrt{10+x}+\sqrt{11-5 x}=-1$

Choose the one alternative that best completes the statement or answers the question. Simplify. Assume that all variables represent positive real numbers. - $-17 \sqrt{14}-10 \sqrt{14}$

Choose the one alternative that best completes the statement or answers the question. Find the root if it is a real number. - $\sqrt[4]{-625}$

Rewrite the expressions with rational exponents as radical expressions, and then solve the equation. - $(10+4 t)^{1 / 3}-(7+6 t)^{1 / 3}=0$

Review of the Real Number System

Linear Equations, Inequalities, and Applications

Linear Equations, Graphs, and Functions

Systems of Linear Equations

Exponents, Polynomials, and Polynomial Functions

Factoring

Rational Expressions and Functions

Quadratic Equations, Inequalities, and Functions

Inverse, Exponential, and Logarithmic Functions

Nonlinear Functions, Conic Sections, and Nonlinear Systems

Further Topics in Algebra

Appendices

Filters

Exam 8: Roots, Radicals, and Root Functions

Multiply, then simplify the product. Assume that all variables represent positive real numbers. - (5x+5)(3x+3)(\sqrt{5 x}+5)(\sqrt{3 x}+3)(5x​+5)(3x​+3)

Simplify. Assume that all variables represent positive real numbers. - 500k7q8\sqrt{500 k^{7} q^{8}}500k7q8​

Choose the one alternative that best completes the statement or answers the question. Solve the equation - 7x−2−9=0\sqrt{7 \mathrm{x}-2}-9=07x−2​−9=0

(9−4i)2(9-4 \mathrm{i})^{2}(9−4i)2

Use a calculator to approximate the root to the nearest thousandth. - 251\sqrt{251}251​

Find the decimal approximation for the radical. Round the answer to three decimal places. - 673\sqrt[3]{67}367​

Choose the one alternative that best completes the statement or answers the question. Solve the equation - x−44+10=0\sqrt[4]{\mathrm{x}-4}+10=04x−4​+10=0

Write the expression in the form a + bi. - 6+2i9−3i\frac{6+2 i}{9-3 i}9−3i6+2i​

Simplify. Assume that all variables represent positive real numbers. - 16y4\sqrt[4]{\frac{16}{\mathrm{y}}}4y16​​

Simplify the radical. Assume that all variables represent positive real numbers - 6y77776y34\sqrt[4]{\frac{6 y^{7}}{7776 y^{3}}}47776y36y7​​

Find the distance between the pair of points. - (22,211)(2 \sqrt{2}, 2 \sqrt{11})(22​,211​) and (92,−211)(9 \sqrt{2},-2 \sqrt{11})(92​,−211​)

Simplify the expression involving rational exponents. - (−8)2/3(-8)^{2 / 3}(−8)2/3

Choose the one alternative that best completes the statement or answers the question. Find the root if it is a real number. - −2435\sqrt[5]{-243}5−243​

Solve the formula for the indicated variable. - T=2πLKT=2 \pi \sqrt{\frac{L}{K}}T=2πKL​​ for LLL

Simplify by first converting to rational exponents. Assume that all variables represent positive real numbers. - 5123\sqrt[3]{5^{12}}3512​

Choose the one alternative that best completes the statement or answers the question. Solve the equation - 10+x+11−5x=−1\sqrt{10+x}+\sqrt{11-5 x}=-110+x​+11−5x​=−1

Choose the one alternative that best completes the statement or answers the question. Simplify. Assume that all variables represent positive real numbers. - −1714−1014-17 \sqrt{14}-10 \sqrt{14}−1714​−1014​

Choose the one alternative that best completes the statement or answers the question. Find the root if it is a real number. - −6254\sqrt[4]{-625}4−625​

Rewrite the expressions with rational exponents as radical expressions, and then solve the equation. - (10+4t)1/3−(7+6t)1/3=0(10+4 t)^{1 / 3}-(7+6 t)^{1 / 3}=0(10+4t)1/3−(7+6t)1/3=0

Review of the Real Number System

Linear Equations, Inequalities, and Applications

Linear Equations, Graphs, and Functions

Systems of Linear Equations

Exponents, Polynomials, and Polynomial Functions

Factoring

Rational Expressions and Functions

Quadratic Equations, Inequalities, and Functions

Inverse, Exponential, and Logarithmic Functions

Nonlinear Functions, Conic Sections, and Nonlinear Systems

Further Topics in Algebra

Appendices

Filters

Multiply, then simplify the product. Assume that all variables represent positive real numbers. - $(\sqrt{5 x}+5)(\sqrt{3 x}+3)$

Simplify. Assume that all variables represent positive real numbers. - $\sqrt{500 k^{7} q^{8}}$

Choose the one alternative that best completes the statement or answers the question. Solve the equation - $\sqrt{7 \mathrm{x}-2}-9=0$

$(9-4 \mathrm{i})^{2}$

Use a calculator to approximate the root to the nearest thousandth. - $\sqrt{251}$

Find the decimal approximation for the radical. Round the answer to three decimal places. - $\sqrt[3]{67}$

Choose the one alternative that best completes the statement or answers the question. Solve the equation - $\sqrt[4]{\mathrm{x}-4}+10=0$

Write the expression in the form a + bi. - $\frac{6+2 i}{9-3 i}$

Simplify. Assume that all variables represent positive real numbers. - $\sqrt[4]{\frac{16}{\mathrm{y}}}$

Simplify the radical. Assume that all variables represent positive real numbers - $\sqrt[4]{\frac{6 y^{7}}{7776 y^{3}}}$

Find the distance between the pair of points. - $(2 \sqrt{2}, 2 \sqrt{11})$ and $(9 \sqrt{2},-2 \sqrt{11})$

Simplify the expression involving rational exponents. - $(-8)^{2 / 3}$

Choose the one alternative that best completes the statement or answers the question. Find the root if it is a real number. - $\sqrt[5]{-243}$

Solve the formula for the indicated variable. - $T=2 \pi \sqrt{\frac{L}{K}}$ for $L$

Simplify by first converting to rational exponents. Assume that all variables represent positive real numbers. - $\sqrt[3]{5^{12}}$

Choose the one alternative that best completes the statement or answers the question. Solve the equation - $\sqrt{10+x}+\sqrt{11-5 x}=-1$

Choose the one alternative that best completes the statement or answers the question. Simplify. Assume that all variables represent positive real numbers. - $-17 \sqrt{14}-10 \sqrt{14}$

Choose the one alternative that best completes the statement or answers the question. Find the root if it is a real number. - $\sqrt[4]{-625}$

Rewrite the expressions with rational exponents as radical expressions, and then solve the equation. - $(10+4 t)^{1 / 3}-(7+6 t)^{1 / 3}=0$