Question 1

How many real number cube roots does any negative number have?

Accepted Answer

The answer of How many real number cube roots does...

Question 2

The easiest way to simplify the expression $\sqrt{\frac{7}{27}}$ is to multiply both the numerator and denominator of $\frac{\sqrt{7}}{\sqrt{27}}$ by $\sqrt{27}$.

Accepted Answer

A) True 
 B)False

Question 3

Choose values for $n$, $a$, and $b$ to demonstrate the truth or falsity of the statement:
$\sqrt[n]{a}+\sqrt[n]{b}=\sqrt[n]{a+b}$

Accepted Answer

A) True 
 B)False

Question 4

Use the rules of exponents to simplify the expression. Write the answer with positive exponents. Assume that all variables represent positive real numbers.
-$\left(16 \frac{a^{3} b^{-4}}{a^{-1} b^{4}}\right)^{1 / 4}$

Accepted Answer

A)  $2 a b^{2}$ 
B)  $\frac{4 a}{b^{2}}$ 
C)  $\frac{2 \mathrm{a}}{\mathrm{b}^{2}}$ 
D)  $\frac{4 b^{2}}{a}$ 
A)  $2 a b^{2}$ 
B)  $\frac{4 a}{b^{2}}$ 
C)  $\frac{2 \mathrm{a}}{\mathrm{b}^{2}}$ 
D)  $\frac{4 b^{2}}{a}$

Question 5

Multiply, then simplify the product. Assume that all variables represent positive real numbers.
-$(4 \sqrt{x}+\sqrt{y})(4 \sqrt{x}-\sqrt{y})$

Accepted Answer

A)  $16 x-y$ 
B)  $16 x^{2}-y^{2}$ 
C)  $16 x-2 \sqrt{x y}-y$ 
D)  $16 x+y$ 
A)  $16 x-y$ 
B)  $16 x^{2}-y^{2}$ 
C)  $16 x-2 \sqrt{x y}-y$ 
D)  $16 x+y$

Question 6

Simplify. Assume that all variables represent positive real numbers.
-$\sqrt[3]{64 \mathrm{a}^{8} \mathrm{~b}^{5}}$

Accepted Answer

A)  $4 a b \sqrt[3]{a^{2} b^{2}}$ 
B)  $4 \sqrt[3]{\mathrm{a}^{2} \mathrm{~b}^{2}}$ 
C)  $4 a^{2} b \sqrt[3]{a^{2} b^{2}}$ 
D)  $4 a b \sqrt[3]{a^{3} b^{3}}$ 
A)  $4 a b \sqrt[3]{a^{2} b^{2}}$ 
B)  $4 \sqrt[3]{\mathrm{a}^{2} \mathrm{~b}^{2}}$ 
C)  $4 a^{2} b \sqrt[3]{a^{2} b^{2}}$ 
D)  $4 a b \sqrt[3]{a^{3} b^{3}}$

Question 7

Choose the one alternative that best completes the statement or answers the question.
Solve the equation
-$\sqrt{2 x+3}+\sqrt{4-x}=4$

Accepted Answer

A)  $\varnothing$ 
B)  $\left\{3, \frac{11}{9}\right\}$ 
C)  $\{3\}$ 
D)  $\{-3\}$ 
A)  $\varnothing$ 
B)  $\left\{3, \frac{11}{9}\right\}$ 
C)  $\{3\}$ 
D)  $\{-3\}$

Question 8

Choose the one alternative that best completes the statement or answers the question.
Simplify. Assume that all variables represent positive real numbers.
-$\sqrt{6 x}-\frac{\sqrt{x}}{\sqrt{6}}$

Accepted Answer

A)  0 
B)  $\frac{\sqrt{6 x}}{6}$ 
C)  $\frac{5 \sqrt{6 x}}{6}$ 
D)  $-\frac{1}{6}$ 
A)  0 
B)  $\frac{\sqrt{6 x}}{6}$ 
C)  $\frac{5 \sqrt{6 x}}{6}$ 
D)  $-\frac{1}{6}$

Question 9

Rewrite the expressions with rational exponents as radical expressions, and then solve the equation.
-$(3 x+1)^{1 / 2}=3+(x-4)^{1 / 2}$

Accepted Answer

A)  $\{-5,-8\}$ 
B)  $\varnothing$ 
C)  $\{-1\}$ 
D)  $\{5,8\}$ 
A)  $\{-5,-8\}$ 
B)  $\varnothing$ 
C)  $\{-1\}$ 
D)  $\{5,8\}$

Question 10

Write the number as a product of a real number and $i$. Simplify the radical expression.
-$\sqrt{-289}$

Accepted Answer

A)  $-17 \mathrm{i}$ 
B)  $-\mathrm{i} \sqrt{17}$ 
C)  $17 \mathrm{i}$ 
D)  ± 17 
A)  $-17 \mathrm{i}$ 
B)  $-\mathrm{i} \sqrt{17}$ 
C)  $17 \mathrm{i}$ 
D)  ± 17

Question 11

$\mathrm{i}(9-3 \mathrm{i})(2-9 \mathrm{i})$

Accepted Answer

A)  $27 \mathrm{i}^{3}+87 \mathrm{i}^{2}+18 \mathrm{i}$ 
B)  $45+75 i$ 
C)  $87-9 \mathrm{i}$ 
D)  $-87+9 i$ 
A)  $27 \mathrm{i}^{3}+87 \mathrm{i}^{2}+18 \mathrm{i}$ 
B)  $45+75 i$ 
C)  $87-9 \mathrm{i}$ 
D)  $-87+9 i$

Question 12

Choose the one alternative that best completes the statement or answers the question.
Simplify. Assume that all variables represent positive real numbers.
-$6 \sqrt[4]{m^{9} p^{6}}-2 m^{2} p \sqrt[4]{m p^{2}}$

Accepted Answer

A)  4 
B)  $4 m^{2} p \sqrt[4]{m p^{2}}$ 
C)  Cannot be simplified 
D)  $8 m p^{2} \sqrt[4]{m p^{2}}$ 
A)  4 
B)  $4 m^{2} p \sqrt[4]{m p^{2}}$ 
C)  Cannot be simplified 
D)  $8 m p^{2} \sqrt[4]{m p^{2}}$

Question 13

Simplify the root.
-$-\sqrt[4]{\mathrm{x}^{28}}$

Accepted Answer

A)  $x^{7}$ 
B)  $-\left|x^{7}\right|$ 
C)  $-x^{7}$ 
D)  $\left|-x^{7}\right|$ 
A)  $x^{7}$ 
B)  $-\left|x^{7}\right|$ 
C)  $-x^{7}$ 
D)  $\left|-x^{7}\right|$

Question 14

Simplify the root.
-$\sqrt[3]{\mathrm{x}^{3}}$

Accepted Answer

A)  $x$ 
B)  $|-x|$ 
C) $-x$ 
D) $ |x| $ 
A)  $x$ 
B)  $|-x|$ 
C) $-x$ 
D) $ |x| $

Question 15

Simplify the expression involving rational exponents.
-$77764 / 5$

Accepted Answer

A)  1,679,616 
B)  1296 
C)  279,936 
D)  10,077,696 
A)  1,679,616 
B)  1296 
C)  279,936 
D)  10,077,696

Question 16

Simplify the radical. Assume that all variables represent positive real numbers
-$\begin{aligned} &-\sqrt[4]{\frac{16}{m^{4}}} \end{aligned}$

Accepted Answer

A) $-\frac{2}{m} $ 
B)  Not a real number 
C)  } \frac{4}{m^{2}} 
D)  $\frac{2}{m}$ 
A) $-\frac{2}{m} $ 
B)  Not a real number 
C)  } \frac{4}{m^{2}} 
D)  $\frac{2}{m}$

Question 17

Use the rules of exponents to simplify the expression. Write the answer with positive exponents. Assume that all variables represent positive real numbers.
-$\left(b^{4}\right)^{5 / 4}$

Accepted Answer

A)  $b^{5 / 16}$ 
B)  $b^{9 / 4}$ 
C)  $b^{5}$ 
D)  $\mathrm{b}^{1 / 4}$ 
A)  $b^{5 / 16}$ 
B)  $b^{9 / 4}$ 
C)  $b^{5}$ 
D)  $\mathrm{b}^{1 / 4}$

Question 18

Rationalize the denominator. Assume that all variables represent positive real numbers and that the denominator is not zero.
-$\frac{\sqrt{3}}{\sqrt{7}+2}$

Accepted Answer

A)  $\frac{\sqrt{21}-2 \sqrt{3}}{3}$ 
B)  $\frac{3 \sqrt{21}+73}{14}$ 
C)  $\frac{\sqrt{21}+2 \sqrt{3}}{3}$ 
D)  $\frac{\sqrt{21}-2 \sqrt{3}}{9}$ 
A)  $\frac{\sqrt{21}-2 \sqrt{3}}{3}$ 
B)  $\frac{3 \sqrt{21}+73}{14}$ 
C)  $\frac{\sqrt{21}+2 \sqrt{3}}{3}$ 
D)  $\frac{\sqrt{21}-2 \sqrt{3}}{9}$

Question 19

Choose the one alternative that best completes the statement or answers the question.
Find the root if it is a real number.
-$-\sqrt[5]{\frac{1}{1024}}$

Accepted Answer

A)  $-\frac{1}{4}$ 
B)  $-\frac{5}{1024}$ 
C)  -4 
D)  $\varnothing$ 
A)  $-\frac{1}{4}$ 
B)  $-\frac{5}{1024}$ 
C)  -4 
D)  $\varnothing$

Question 20

Express the radical in simplified form.Assume that all variables represent positive real numbers.
-$\sqrt{150}$

Accepted Answer

A)  $25 \sqrt{6}$ 
B)  5 
C)  12 
D)  $5 \sqrt{6}$ 
A)  $25 \sqrt{6}$ 
B)  5 
C)  12 
D)  $5 \sqrt{6}$

How many real number cube roots does any negative number have?

The easiest way to simplify the expression $\sqrt{\frac{7}{27}}$ is to multiply both the numerator and denominator of $\frac{\sqrt{7}}{\sqrt{27}}$ by $\sqrt{27}$ .

Choose values for $n$ , $a$ , and $b$ to demonstrate the truth or falsity of the statement: $\sqrt[n]{a}+\sqrt[n]{b}=\sqrt[n]{a+b}$

Use the rules of exponents to simplify the expression. Write the answer with positive exponents. Assume that all variables represent positive real numbers. - $\left(16 \frac{a^{3} b^{-4}}{a^{-1} b^{4}}\right)^{1 / 4}$

Multiply, then simplify the product. Assume that all variables represent positive real numbers. - $(4 \sqrt{x}+\sqrt{y})(4 \sqrt{x}-\sqrt{y})$

Simplify. Assume that all variables represent positive real numbers. - $\sqrt[3]{64 \mathrm{a}^{8} \mathrm{~b}^{5}}$

Choose the one alternative that best completes the statement or answers the question. Solve the equation - $\sqrt{2 x+3}+\sqrt{4-x}=4$

Choose the one alternative that best completes the statement or answers the question. Simplify. Assume that all variables represent positive real numbers. - $\sqrt{6 x}-\frac{\sqrt{x}}{\sqrt{6}}$

Rewrite the expressions with rational exponents as radical expressions, and then solve the equation. - $(3 x+1)^{1 / 2}=3+(x-4)^{1 / 2}$

Write the number as a product of a real number and $i$ . Simplify the radical expression. - $\sqrt{-289}$

$\mathrm{i}(9-3 \mathrm{i})(2-9 \mathrm{i})$

Choose the one alternative that best completes the statement or answers the question. Simplify. Assume that all variables represent positive real numbers. - $6 \sqrt[4]{m^{9} p^{6}}-2 m^{2} p \sqrt[4]{m p^{2}}$

Simplify the root. - $-\sqrt[4]{\mathrm{x}^{28}}$

Simplify the root. - $\sqrt[3]{\mathrm{x}^{3}}$

Simplify the expression involving rational exponents. - $77764 / 5$

Simplify the radical. Assume that all variables represent positive real numbers - -

Use the rules of exponents to simplify the expression. Write the answer with positive exponents. Assume that all variables represent positive real numbers. - $\left(b^{4}\right)^{5 / 4}$

Rationalize the denominator. Assume that all variables represent positive real numbers and that the denominator is not zero. - $\frac{\sqrt{3}}{\sqrt{7}+2}$

Choose the one alternative that best completes the statement or answers the question. Find the root if it is a real number. - $-\sqrt[5]{\frac{1}{1024}}$

Express the radical in simplified form.Assume that all variables represent positive real numbers. - $\sqrt{150}$

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

How many real number cube roots does any negative number have?

The easiest way to simplify the expression 727\sqrt{\frac{7}{27}}277​​ is to multiply both the numerator and denominator of 727\frac{\sqrt{7}}{\sqrt{27}}27​7​​ by 27\sqrt{27}27​ .

Choose values for nnn , aaa , and bbb to demonstrate the truth or falsity of the statement: an+bn=a+bn\sqrt[n]{a}+\sqrt[n]{b}=\sqrt[n]{a+b}na​+nb​=na+b​

Use the rules of exponents to simplify the expression. Write the answer with positive exponents. Assume that all variables represent positive real numbers. - (16a3b−4a−1b4)1/4\left(16 \frac{a^{3} b^{-4}}{a^{-1} b^{4}}\right)^{1 / 4}(16a−1b4a3b−4​)1/4

Multiply, then simplify the product. Assume that all variables represent positive real numbers. - (4x+y)(4x−y)(4 \sqrt{x}+\sqrt{y})(4 \sqrt{x}-\sqrt{y})(4x​+y​)(4x​−y​)

Simplify. Assume that all variables represent positive real numbers. - 64a8 b53\sqrt[3]{64 \mathrm{a}^{8} \mathrm{~b}^{5}}364a8 b5​

Choose the one alternative that best completes the statement or answers the question. Solve the equation - 2x+3+4−x=4\sqrt{2 x+3}+\sqrt{4-x}=42x+3​+4−x​=4

Choose the one alternative that best completes the statement or answers the question. Simplify. Assume that all variables represent positive real numbers. - 6x−x6\sqrt{6 x}-\frac{\sqrt{x}}{\sqrt{6}}6x​−6​x​​

Rewrite the expressions with rational exponents as radical expressions, and then solve the equation. - (3x+1)1/2=3+(x−4)1/2(3 x+1)^{1 / 2}=3+(x-4)^{1 / 2}(3x+1)1/2=3+(x−4)1/2

Write the number as a product of a real number and iii . Simplify the radical expression. - −289\sqrt{-289}−289​

i(9−3i)(2−9i)\mathrm{i}(9-3 \mathrm{i})(2-9 \mathrm{i})i(9−3i)(2−9i)

Choose the one alternative that best completes the statement or answers the question. Simplify. Assume that all variables represent positive real numbers. - 6m9p64−2m2pmp246 \sqrt[4]{m^{9} p^{6}}-2 m^{2} p \sqrt[4]{m p^{2}}64m9p6​−2m2p4mp2​

Simplify the root. - −x284-\sqrt[4]{\mathrm{x}^{28}}−4x28​

Simplify the root. - x33\sqrt[3]{\mathrm{x}^{3}}3x3​

Simplify the expression involving rational exponents. - 77764/577764 / 577764/5

Simplify the radical. Assume that all variables represent positive real numbers - -

Use the rules of exponents to simplify the expression. Write the answer with positive exponents. Assume that all variables represent positive real numbers. - (b4)5/4\left(b^{4}\right)^{5 / 4}(b4)5/4

Rationalize the denominator. Assume that all variables represent positive real numbers and that the denominator is not zero. - 37+2\frac{\sqrt{3}}{\sqrt{7}+2}7​+23​​

Choose the one alternative that best completes the statement or answers the question. Find the root if it is a real number. - −110245-\sqrt[5]{\frac{1}{1024}}−510241​​

Express the radical in simplified form.Assume that all variables represent positive real numbers. - 150\sqrt{150}150​

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

The easiest way to simplify the expression $\sqrt{\frac{7}{27}}$ is to multiply both the numerator and denominator of $\frac{\sqrt{7}}{\sqrt{27}}$ by $\sqrt{27}$ .

Choose values for $n$ , $a$ , and $b$ to demonstrate the truth or falsity of the statement: $\sqrt[n]{a}+\sqrt[n]{b}=\sqrt[n]{a+b}$

Use the rules of exponents to simplify the expression. Write the answer with positive exponents. Assume that all variables represent positive real numbers. - $\left(16 \frac{a^{3} b^{-4}}{a^{-1} b^{4}}\right)^{1 / 4}$

Multiply, then simplify the product. Assume that all variables represent positive real numbers. - $(4 \sqrt{x}+\sqrt{y})(4 \sqrt{x}-\sqrt{y})$

Simplify. Assume that all variables represent positive real numbers. - $\sqrt[3]{64 \mathrm{a}^{8} \mathrm{~b}^{5}}$

Choose the one alternative that best completes the statement or answers the question. Solve the equation - $\sqrt{2 x+3}+\sqrt{4-x}=4$

Choose the one alternative that best completes the statement or answers the question. Simplify. Assume that all variables represent positive real numbers. - $\sqrt{6 x}-\frac{\sqrt{x}}{\sqrt{6}}$

Rewrite the expressions with rational exponents as radical expressions, and then solve the equation. - $(3 x+1)^{1 / 2}=3+(x-4)^{1 / 2}$

Write the number as a product of a real number and $i$ . Simplify the radical expression. - $\sqrt{-289}$

$\mathrm{i}(9-3 \mathrm{i})(2-9 \mathrm{i})$

Choose the one alternative that best completes the statement or answers the question. Simplify. Assume that all variables represent positive real numbers. - $6 \sqrt[4]{m^{9} p^{6}}-2 m^{2} p \sqrt[4]{m p^{2}}$

Simplify the root. - $-\sqrt[4]{\mathrm{x}^{28}}$

Simplify the root. - $\sqrt[3]{\mathrm{x}^{3}}$

Simplify the expression involving rational exponents. - $77764 / 5$

Use the rules of exponents to simplify the expression. Write the answer with positive exponents. Assume that all variables represent positive real numbers. - $\left(b^{4}\right)^{5 / 4}$

Rationalize the denominator. Assume that all variables represent positive real numbers and that the denominator is not zero. - $\frac{\sqrt{3}}{\sqrt{7}+2}$

Choose the one alternative that best completes the statement or answers the question. Find the root if it is a real number. - $-\sqrt[5]{\frac{1}{1024}}$

Express the radical in simplified form.Assume that all variables represent positive real numbers. - $\sqrt{150}$