Question 1

Can $\sqrt{a}$ represent a negative real number?

Accepted Answer

The answer of Can $\sqrt{a}$ represent a negative real number?...

Question 2

Rationalize the denominator. Assume that all variables represent positive real numbers and that the denominator is not zero.
-$\frac{\sqrt{2}}{2 \sqrt{7}-\sqrt{2}}$

Accepted Answer

A)  $\frac{\sqrt{14}-1}{13}$ 
B)  $\frac{\sqrt{14}+1}{13}$ 
C)  $\frac{\sqrt{14}+1}{15}$ 
D)  $\frac{\sqrt{7}+1}{13}$ 
A)  $\frac{\sqrt{14}-1}{13}$ 
B)  $\frac{\sqrt{14}+1}{13}$ 
C)  $\frac{\sqrt{14}+1}{15}$ 
D)  $\frac{\sqrt{7}+1}{13}$

Question 3

Rationalize the denominator. Assume that all variables represent positive real numbers. 
-$-\sqrt{\frac{49}{8}}$

Accepted Answer

A)  $-\frac{7 \sqrt{2}}{2}$ 
B)  $-\frac{7 \sqrt{2}}{4}$ 
C)  -8 
D)  $-7 \sqrt{2}$ 
A)  $-\frac{7 \sqrt{2}}{2}$ 
B)  $-\frac{7 \sqrt{2}}{4}$ 
C)  -8 
D)  $-7 \sqrt{2}$

Question 4

Simplify by first writing the radicals with the same index. Then multiply.
-$\sqrt{3} \cdot \sqrt[3]{4}$

Accepted Answer

A)  $\sqrt[6]{144}$ 
B)  $\sqrt[6]{108}$ 
C)  $\sqrt[6]{12}$ 
D)  $\sqrt[6]{432}$ 
A)  $\sqrt[6]{144}$ 
B)  $\sqrt[6]{108}$ 
C)  $\sqrt[6]{12}$ 
D)  $\sqrt[6]{432}$

Question 5

Choose the one alternative that best completes the statement or answers the question.
Multiply using the product rule.
-$\sqrt[3]{9 x} \cdot \sqrt[3]{49 x}$

Accepted Answer

A)  $10 x$ 
B)  $21 x$ 
C)  $\sqrt[3]{58 x^{2}}$ 
D)  $\sqrt[3]{441 x^{2}}$ 
A)  $10 x$ 
B)  $21 x$ 
C)  $\sqrt[3]{58 x^{2}}$ 
D)  $\sqrt[3]{441 x^{2}}$

Question 6

Find the distance between the pair of points.
-$(m, m+n)$ and $(m-n, n)$

Accepted Answer

A)  $n-m$ 
B)  $\sqrt{n^{2}+m^{2}}$ 
C)  $\sqrt{n^{2}-m^{2}}$ 
D)  $n+m$ 
A)  $n-m$ 
B)  $\sqrt{n^{2}+m^{2}}$ 
C)  $\sqrt{n^{2}-m^{2}}$ 
D)  $n+m$

Question 7

Suppose a rectangle has length $\sqrt{85}$ and width $\sqrt{99}$. Which of the following is the best estimate of its area?

Accepted Answer

A)  172 
B)  28 
C)  38 
D)  90 
A)  172 
B)  28 
C)  38 
D)  90

Question 8

Solve the equation.
-$5 x+\sqrt{x+56}=8+3 x$

Accepted Answer

A)  $\{64\}$ 
B)  $\{16\}$ 
C)  $\left\{8, \frac{1}{4}\right\}$ 
D)  $\left\{-8,-\frac{1}{4}\right\}$ 
A)  $\{64\}$ 
B)  $\{16\}$ 
C)  $\left\{8, \frac{1}{4}\right\}$ 
D)  $\left\{-8,-\frac{1}{4}\right\}$

Question 9

Find the distance between the pair of points.
-$(-1,-2)$ and $(1,-6)$

Accepted Answer

A)  $2 \sqrt{5}$ 
B)  6 
C)  $2 \sqrt{3}$ 
D)  18 
A)  $2 \sqrt{5}$ 
B)  6 
C)  $2 \sqrt{3}$ 
D)  18

Question 10

(6i)(2i)

Accepted Answer

A)  8 
B)  -8 
C)  -12 
D)  12 
A)  8 
B)  -8 
C)  -12 
D)  12

Question 11

Choose the one alternative that best completes the statement or answers the question.
Solve the equation
-$\sqrt{x+3}=x-3$

Accepted Answer

A)  $\{1,6\}$ 
B)  $\{6,13\}$ 
C)  $\{6\}$ 
D)  $\{1,13\}$ 
A)  $\{1,6\}$ 
B)  $\{6,13\}$ 
C)  $\{6\}$ 
D)  $\{1,13\}$

Question 12

Multiply or divide as indicated.
-$\frac{\sqrt{-1000}}{\sqrt{-10}}$

Accepted Answer

A)  $-10 \mathrm{i}$ 
B)  -10 
C)  10 
D)  $10 \mathrm{i}$ 
A)  $-10 \mathrm{i}$ 
B)  -10 
C)  10 
D)  $10 \mathrm{i}$

Question 13

Choose the one alternative that best completes the statement or answers the question.
Solve the equation
-$\sqrt{x^{2}+2 x+22}+4=x$

Accepted Answer

A)  $\left\{\frac{3}{5}\right\}$ 
B)  $\left\{-\frac{3}{5}\right\}$ 
C)  $\varnothing$ 
D)  $\left\{\frac{5}{3}\right\}$ 
A)  $\left\{\frac{3}{5}\right\}$ 
B)  $\left\{-\frac{3}{5}\right\}$ 
C)  $\varnothing$ 
D)  $\left\{\frac{5}{3}\right\}$

Question 14

$(6-5 i)(2+9 i)$

Accepted Answer

A)  $-45 i^{2}+44 i+12$ 
B)  $57-44 \mathrm{i}$ 
C)  $57+44 \mathrm{i}$ 
D)  $-33-64 \mathrm{i}$ 
A)  $-45 i^{2}+44 i+12$ 
B)  $57-44 \mathrm{i}$ 
C)  $57+44 \mathrm{i}$ 
D)  $-33-64 \mathrm{i}$

Question 15

$(7+5 i)(8-8 i)$

Accepted Answer

A)  $96+16 i$ 
B)  $16+96 i$ 
C)  $-40 i^{2}-16 i+56$ 
D)  $96-16 \mathrm{i}$ 
A)  $96+16 i$ 
B)  $16+96 i$ 
C)  $-40 i^{2}-16 i+56$ 
D)  $96-16 \mathrm{i}$

Question 16

Find the decimal approximation for the radical. Round the answer to three decimal places.
-$\sqrt[4]{113}$

Accepted Answer

A)  3.282 
B)  3.292 
C)  3.274 
D)  3.260 
A)  3.282 
B)  3.292 
C)  3.274 
D)  3.260

Question 17

Simplify by first converting to rational exponents. Assume that all variables represent positive real numbers.
-$\sqrt[3]{x^{21}}$

Accepted Answer

A)  $\frac{z}{3}$ 
B)  $x^{63}$ 
C)  $x^{7}$ 
D)  $3 z$ 
A)  $\frac{z}{3}$ 
B)  $x^{63}$ 
C)  $x^{7}$ 
D)  $3 z$

Question 18

Provide an appropriate response.
-$\mathrm{i}^{3}=-\mathrm{i}$

Accepted Answer

A) True 
 B)False

Question 19

Write the number as a product of a real number and $i$. Simplify the radical expression.
-$-\sqrt{-288}$

Accepted Answer

A)  $12 \sqrt{2}$ 
B)  $12 \mathrm{i} \sqrt{2}$ 
C)  $-12 \sqrt{2}$ 
D)  $-12 \mathrm{i} \sqrt{2}$ 
A)  $12 \sqrt{2}$ 
B)  $12 \mathrm{i} \sqrt{2}$ 
C)  $-12 \sqrt{2}$ 
D)  $-12 \mathrm{i} \sqrt{2}$

Question 20

The radius of a right circular cone is given by the formula $r=\sqrt{\frac{3 V}{\pi h}}$, where $V$ is the volume and $h$ is the height. If the volume is 1302 cubic inches and the height is 14 inches, what is the radius? Use 3.14 for $\pi$, and round your answer to the nearest tenth of an inch.

Accepted Answer

A)  0.1 inches 
B)  5.4 inches 
C)  9.9 inches 
D)  9.4 inches 
A)  0.1 inches 
B)  5.4 inches 
C)  9.9 inches 
D)  9.4 inches

Can $\sqrt{a}$ represent a negative real number?

Rationalize the denominator. Assume that all variables represent positive real numbers and that the denominator is not zero. - $\frac{\sqrt{2}}{2 \sqrt{7}-\sqrt{2}}$

Rationalize the denominator. Assume that all variables represent positive real numbers. - $-\sqrt{\frac{49}{8}}$

Simplify by first writing the radicals with the same index. Then multiply. - $\sqrt{3} \cdot \sqrt[3]{4}$

Choose the one alternative that best completes the statement or answers the question. Multiply using the product rule. - $\sqrt[3]{9 x} \cdot \sqrt[3]{49 x}$

Find the distance between the pair of points. - $(m, m+n)$ and $(m-n, n)$

Suppose a rectangle has length $\sqrt{85}$ and width $\sqrt{99}$ . Which of the following is the best estimate of its area?

Solve the equation. - $5 x+\sqrt{x+56}=8+3 x$

Find the distance between the pair of points. - $(-1,-2)$ and $(1,-6)$

(6i)(2i)

Choose the one alternative that best completes the statement or answers the question. Solve the equation - $\sqrt{x+3}=x-3$

Multiply or divide as indicated. - $\frac{\sqrt{-1000}}{\sqrt{-10}}$

Choose the one alternative that best completes the statement or answers the question. Solve the equation - $\sqrt{x^{2}+2 x+22}+4=x$

$(6-5 i)(2+9 i)$

$(7+5 i)(8-8 i)$

Find the decimal approximation for the radical. Round the answer to three decimal places. - $\sqrt[4]{113}$

Simplify by first converting to rational exponents. Assume that all variables represent positive real numbers. - $\sqrt[3]{x^{21}}$

Provide an appropriate response. - $\mathrm{i}^{3}=-\mathrm{i}$

Write the number as a product of a real number and $i$ . Simplify the radical expression. - $-\sqrt{-288}$

The radius of a right circular cone is given by the formula $r=\sqrt{\frac{3 V}{\pi h}}$ , where $V$ is the volume and $h$ is the height. If the volume is 1302 cubic inches and the height is 14 inches, what is the radius? Use 3.14 for $\pi$ , and round your answer to the nearest tenth of an inch.

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

Can a\sqrt{a}a​ represent a negative real number?

Rationalize the denominator. Assume that all variables represent positive real numbers and that the denominator is not zero. - 227−2\frac{\sqrt{2}}{2 \sqrt{7}-\sqrt{2}}27​−2​2​​

Rationalize the denominator. Assume that all variables represent positive real numbers. - −498-\sqrt{\frac{49}{8}}−849​​

Simplify by first writing the radicals with the same index. Then multiply. - 3⋅43\sqrt{3} \cdot \sqrt[3]{4}3​⋅34​

Choose the one alternative that best completes the statement or answers the question. Multiply using the product rule. - 9x3⋅49x3\sqrt[3]{9 x} \cdot \sqrt[3]{49 x}39x​⋅349x​

Find the distance between the pair of points. - (m,m+n)(m, m+n)(m,m+n) and (m−n,n)(m-n, n)(m−n,n)

Suppose a rectangle has length 85\sqrt{85}85​ and width 99\sqrt{99}99​ . Which of the following is the best estimate of its area?

Solve the equation. - 5x+x+56=8+3x5 x+\sqrt{x+56}=8+3 x5x+x+56​=8+3x

Find the distance between the pair of points. - (−1,−2)(-1,-2)(−1,−2) and (1,−6)(1,-6)(1,−6)

(6i)(2i)

Choose the one alternative that best completes the statement or answers the question. Solve the equation - x+3=x−3\sqrt{x+3}=x-3x+3​=x−3

Multiply or divide as indicated. - −1000−10\frac{\sqrt{-1000}}{\sqrt{-10}}−10​−1000​​

Choose the one alternative that best completes the statement or answers the question. Solve the equation - x2+2x+22+4=x\sqrt{x^{2}+2 x+22}+4=xx2+2x+22​+4=x

(6−5i)(2+9i)(6-5 i)(2+9 i)(6−5i)(2+9i)

(7+5i)(8−8i)(7+5 i)(8-8 i)(7+5i)(8−8i)

Find the decimal approximation for the radical. Round the answer to three decimal places. - 1134\sqrt[4]{113}4113​

Simplify by first converting to rational exponents. Assume that all variables represent positive real numbers. - x213\sqrt[3]{x^{21}}3x21​

Provide an appropriate response. - i3=−i\mathrm{i}^{3}=-\mathrm{i}i3=−i

Write the number as a product of a real number and iii . Simplify the radical expression. - −−288-\sqrt{-288}−−288​

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

Can $\sqrt{a}$ represent a negative real number?

Rationalize the denominator. Assume that all variables represent positive real numbers and that the denominator is not zero. - $\frac{\sqrt{2}}{2 \sqrt{7}-\sqrt{2}}$

Rationalize the denominator. Assume that all variables represent positive real numbers. - $-\sqrt{\frac{49}{8}}$

Simplify by first writing the radicals with the same index. Then multiply. - $\sqrt{3} \cdot \sqrt[3]{4}$

Choose the one alternative that best completes the statement or answers the question. Multiply using the product rule. - $\sqrt[3]{9 x} \cdot \sqrt[3]{49 x}$

Find the distance between the pair of points. - $(m, m+n)$ and $(m-n, n)$

Suppose a rectangle has length $\sqrt{85}$ and width $\sqrt{99}$ . Which of the following is the best estimate of its area?

Solve the equation. - $5 x+\sqrt{x+56}=8+3 x$

Find the distance between the pair of points. - $(-1,-2)$ and $(1,-6)$

Choose the one alternative that best completes the statement or answers the question. Solve the equation - $\sqrt{x+3}=x-3$

Multiply or divide as indicated. - $\frac{\sqrt{-1000}}{\sqrt{-10}}$

Choose the one alternative that best completes the statement or answers the question. Solve the equation - $\sqrt{x^{2}+2 x+22}+4=x$

$(6-5 i)(2+9 i)$

$(7+5 i)(8-8 i)$

Find the decimal approximation for the radical. Round the answer to three decimal places. - $\sqrt[4]{113}$

Simplify by first converting to rational exponents. Assume that all variables represent positive real numbers. - $\sqrt[3]{x^{21}}$

Provide an appropriate response. - $\mathrm{i}^{3}=-\mathrm{i}$

Write the number as a product of a real number and $i$ . Simplify the radical expression. - $-\sqrt{-288}$