Question 1

Write with radicals. Assume that all variables represent positive real numbers
-$(5 x)^{3 / 7}+(4 y)^{3 / 5}$

Accepted Answer

A)  $\left(\sqrt[7]{5}^{3}+(\sqrt[5]{4})^{3}\right.$ 
B)  $\left(\sqrt[7]{5}^{3} x+(\sqrt[5]{4})^{3} y\right.$ 
C)  $(\sqrt[7]{5 x})^{3}+\left(\sqrt[5]{4 y}^{3}\right.$ 
D)  $\left(\sqrt[3]{5 x}^{7}+\left(\sqrt[3]{4 y}^{5}\right.\right.$ 
A)  $\left(\sqrt[7]{5}^{3}+(\sqrt[5]{4})^{3}\right.$ 
B)  $\left(\sqrt[7]{5}^{3} x+(\sqrt[5]{4})^{3} y\right.$ 
C)  $(\sqrt[7]{5 x})^{3}+\left(\sqrt[5]{4 y}^{3}\right.$ 
D)  $\left(\sqrt[3]{5 x}^{7}+\left(\sqrt[3]{4 y}^{5}\right.\right.$

Question 2

Simplify the expression involving rational exponents.
-$\left(\frac{64}{729}\right)^{-2 / 3}$

Accepted Answer

A)  $\frac{81}{16}$ 
B)  $\frac{729}{64}$ 
C)  $\frac{16}{81}$ 
D)  $\frac{64}{729}$ 
A)  $\frac{81}{16}$ 
B)  $\frac{729}{64}$ 
C)  $\frac{16}{81}$ 
D)  $\frac{64}{729}$

Question 3

The negative sixth root of 64 is -2 .

Accepted Answer

A) True 
 B)False

Question 4

Choose the one alternative that best completes the statement or answers the question.
Evaluate
-$-\sqrt{196}$

Accepted Answer

A)  Not a real number 
B)  14 
C)  -98 
D)  -14 
A)  Not a real number 
B)  14 
C)  -98 
D)  -14

Question 5

Write the expression in the form a + bi.
-$\frac{8 \mathrm{i}}{4+9 \mathrm{i}}$

Accepted Answer

A)  $\frac{72}{65}-\frac{32}{65} \mathrm{i}$ 
B)  $\frac{72}{97}+\frac{32}{97} \mathrm{i}$ 
C)  $-\frac{72}{97}+\frac{32}{97} \mathrm{i}$ 
D)  $-\frac{72}{65}-\frac{32}{65} \mathrm{i}$ 
A)  $\frac{72}{65}-\frac{32}{65} \mathrm{i}$ 
B)  $\frac{72}{97}+\frac{32}{97} \mathrm{i}$ 
C)  $-\frac{72}{97}+\frac{32}{97} \mathrm{i}$ 
D)  $-\frac{72}{65}-\frac{32}{65} \mathrm{i}$

Question 6

Is $\sqrt{11}$ simplified?

Accepted Answer

A) True 
 B)False

Question 7

Use the Pythagorean formula to find the exact length of side $b$ in the figure.
-

Accepted Answer

A)  $b=\sqrt{165}$ 
B)  $b=\sqrt{557}$ 
C)  $b=\sqrt{10}$ 
D)  $b=\sqrt{5}$ 
A)  $b=\sqrt{165}$ 
B)  $b=\sqrt{557}$ 
C)  $b=\sqrt{10}$ 
D)  $b=\sqrt{5}$

Question 8

Write with rational exponents and then apply the properties of exponents. Assume that all radicands represent positive real numbers. Give answers in exponential form.
-$\sqrt[5]{\sqrt[4]{w}}$

Accepted Answer

A)  $w^{1 / 20}$ 
B)  $\mathrm{w}^{20}$ 
C)  $\frac{1}{w^{20}}$ 
D)  $20 \mathrm{w}$ 
A)  $w^{1 / 20}$ 
B)  $\mathrm{w}^{20}$ 
C)  $\frac{1}{w^{20}}$ 
D)  $20 \mathrm{w}$

Question 9

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

Accepted Answer

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

Question 10

Simplify the radical. Assume that all variables represent positive real numbers
-$\sqrt[4]{\frac{625}{81}}$

Accepted Answer

A)  $\frac{5}{3}$ 
B)  $\frac{625}{81}$ 
C)  $\frac{125}{27}$ 
D)  $\frac{25}{9}$ 
A)  $\frac{5}{3}$ 
B)  $\frac{625}{81}$ 
C)  $\frac{125}{27}$ 
D)  $\frac{25}{9}$

Question 11

Find the equation of a circle satisfying the given conditions.
-Center: $(6,0)$; radius: 6

Accepted Answer

A)  $x^{2}+(y-6)^{2}=6$ 
B)  $(x+6)^{2}+y^{2}=36$ 
C)  $x^{2}+(y+6)^{2}=6$ 
D)  $(x-6)^{2}+y^{2}=36$ 
A)  $x^{2}+(y-6)^{2}=6$ 
B)  $(x+6)^{2}+y^{2}=36$ 
C)  $x^{2}+(y+6)^{2}=6$ 
D)  $(x-6)^{2}+y^{2}=36$

Question 12

Simplify the expression. Assume that all variables represent positive real numbers.
-$\frac{5^{3 / 4} x^{-2 / 5} y^{3 / 2}}{5^{-5 / 4} x^{13 / 5} y^{1 / 4}}$

Accepted Answer

A)  $\frac{125 y^{5 / 4}}{x^{3}}$ 
B)  $\frac{25 y^{5 / 8}}{x^{4}}$ 
C)  $\frac{25 y^{5 / 4}}{x^{3}}$ 
D)  $\frac{25 y^{5 / 4}}{x^{2}}$ 
A)  $\frac{125 y^{5 / 4}}{x^{3}}$ 
B)  $\frac{25 y^{5 / 8}}{x^{4}}$ 
C)  $\frac{25 y^{5 / 4}}{x^{3}}$ 
D)  $\frac{25 y^{5 / 4}}{x^{2}}$

Question 13

The length of the diagonal of a box is given by $\mathrm{D}=\sqrt{\mathrm{L}^{2}+\mathrm{W}^{2}+\mathrm{H}^{2}}$ where $\mathrm{L}, \mathrm{W}$, and $\mathrm{H}$ are the length, width, and height of the box. Find the length of the diagonal, $\mathrm{D}$, of a box that is $4 \mathrm{ft}$ long, $2 \mathrm{ft}$ high, and $4 \mathrm{ft}$ wide. Give the exact value.

Accepted Answer

A)  $32 \mathrm{ft}$ 
B)  $10 \mathrm{ft}$ 
C)  $4 \sqrt{2} \mathrm{ft}$ 
D)  $6 \mathrm{ft}$ 
A)  $32 \mathrm{ft}$ 
B)  $10 \mathrm{ft}$ 
C)  $4 \sqrt{2} \mathrm{ft}$ 
D)  $6 \mathrm{ft}$

Question 14

Explain why if $\mathrm{n}$ is odd, $\mathrm{x}^{1 / \mathrm{n}}$ is defined even if $\mathrm{x}$ is negative.

Accepted Answer

Answers many vary. One possibi

Question 15

Choose the one alternative that best completes the statement or answers the question.
Solve the equation
-$\sqrt{\mathrm{x}+7}+5=\mathrm{x}$

Accepted Answer

A)  $\{9,18\}$ 
B)  $\{2\}$ 
C)  $\{2,9\}$ 
D)  $\{9\}$ 
A)  $\{9,18\}$ 
B)  $\{2\}$ 
C)  $\{2,9\}$ 
D)  $\{9\}$

Question 16

Find the power of $i$.
-$\mathrm{i}^{3}$

Accepted Answer

A)  -1 
B)  1 
C)  $\mathrm{i}$ 
D)  -i 
A)  -1 
B)  1 
C)  $\mathrm{i}$ 
D)  -i

Question 17

Multiply, then simplify the product. Assume that all variables represent positive real numbers.
-$(3+\sqrt{3})^{2}$

Accepted Answer

A)  $6+6 \sqrt{3}$ 
B)  $12+6 \sqrt{3}$ 
C)  $9+6 \sqrt{3}$ 
D)  $12+3 \sqrt{3}$ 
A)  $6+6 \sqrt{3}$ 
B)  $12+6 \sqrt{3}$ 
C)  $9+6 \sqrt{3}$ 
D)  $12+3 \sqrt{3}$

Question 18

Simplify the radical. Assume that all variables represent positive real numbers.
-$\sqrt[6]{27}$

Accepted Answer

A)  3 
B)  $\sqrt{3}$ 
C)  $\sqrt[3]{3}$ 
D)  $3 \sqrt{3}$ 
A)  3 
B)  $\sqrt{3}$ 
C)  $\sqrt[3]{3}$ 
D)  $3 \sqrt{3}$

Question 19

Solve the problem. Give the answer as a simplified radical expression.
-Find the perimeter of a triangle with sides measuring $2 \sqrt{8}, 4 \sqrt{50}$, and $5 \sqrt{80}$ meters.

Accepted Answer

A)  $24+20 \sqrt{5}$ meters 
B)  $24 \sqrt{2}+20$ meters 
C)  $24 \sqrt{2}+20 \sqrt{5}$ meters 
D)  $44 \sqrt{2}$ meters 
A)  $24+20 \sqrt{5}$ meters 
B)  $24 \sqrt{2}+20$ meters 
C)  $24 \sqrt{2}+20 \sqrt{5}$ meters 
D)  $44 \sqrt{2}$ meters

Question 20

Explain how to evaluate $(-64)^{4 / 3}$.

Accepted Answer

Answers may vary. One possibil

Write with radicals. Assume that all variables represent positive real numbers - $(5 x)^{3 / 7}+(4 y)^{3 / 5}$

Simplify the expression involving rational exponents. - $\left(\frac{64}{729}\right)^{-2 / 3}$

The negative sixth root of 64 is -2 .

Choose the one alternative that best completes the statement or answers the question. Evaluate - $-\sqrt{196}$

Write the expression in the form a + bi. - $\frac{8 \mathrm{i}}{4+9 \mathrm{i}}$

Is $\sqrt{11}$ simplified?

Use the Pythagorean formula to find the exact length of side $b$ in the figure. -

Write with rational exponents and then apply the properties of exponents. Assume that all radicands represent positive real numbers. Give answers in exponential form. - $\sqrt[5]{\sqrt[4]{w}}$

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

Simplify the radical. Assume that all variables represent positive real numbers - $\sqrt[4]{\frac{625}{81}}$

Find the equation of a circle satisfying the given conditions. -Center: $(6,0)$ ; radius: 6

Simplify the expression. Assume that all variables represent positive real numbers. - $\frac{5^{3 / 4} x^{-2 / 5} y^{3 / 2}}{5^{-5 / 4} x^{13 / 5} y^{1 / 4}}$

Explain why if $\mathrm{n}$ is odd, $\mathrm{x}^{1 / \mathrm{n}}$ is defined even if $\mathrm{x}$ is negative.

Choose the one alternative that best completes the statement or answers the question. Solve the equation - $\sqrt{\mathrm{x}+7}+5=\mathrm{x}$

Find the power of $i$ . - $\mathrm{i}^{3}$

Multiply, then simplify the product. Assume that all variables represent positive real numbers. - $(3+\sqrt{3})^{2}$

Simplify the radical. Assume that all variables represent positive real numbers. - $\sqrt[6]{27}$

Solve the problem. Give the answer as a simplified radical expression. -Find the perimeter of a triangle with sides measuring $2 \sqrt{8}, 4 \sqrt{50}$ , and $5 \sqrt{80}$ meters.

Explain how to evaluate $(-64)^{4 / 3}$ .

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

Write with radicals. Assume that all variables represent positive real numbers - (5x)3/7+(4y)3/5(5 x)^{3 / 7}+(4 y)^{3 / 5}(5x)3/7+(4y)3/5

Simplify the expression involving rational exponents. - (64729)−2/3\left(\frac{64}{729}\right)^{-2 / 3}(72964​)−2/3

The negative sixth root of 64 is -2 .

Choose the one alternative that best completes the statement or answers the question. Evaluate - −196-\sqrt{196}−196​

Write the expression in the form a + bi. - 8i4+9i\frac{8 \mathrm{i}}{4+9 \mathrm{i}}4+9i8i​

Is 11\sqrt{11}11​ simplified?

Use the Pythagorean formula to find the exact length of side bbb in the figure. -

Write with rational exponents and then apply the properties of exponents. Assume that all radicands represent positive real numbers. Give answers in exponential form. - w45\sqrt[5]{\sqrt[4]{w}}54w​​

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

Simplify the radical. Assume that all variables represent positive real numbers - 625814\sqrt[4]{\frac{625}{81}}481625​​

Find the equation of a circle satisfying the given conditions. -Center: (6,0)(6,0)(6,0) ; radius: 6

Simplify the expression. Assume that all variables represent positive real numbers. - 53/4x−2/5y3/25−5/4x13/5y1/4\frac{5^{3 / 4} x^{-2 / 5} y^{3 / 2}}{5^{-5 / 4} x^{13 / 5} y^{1 / 4}}5−5/4x13/5y1/453/4x−2/5y3/2​

Explain why if n\mathrm{n}n is odd, x1/n\mathrm{x}^{1 / \mathrm{n}}x1/n is defined even if x\mathrm{x}x is negative.

Choose the one alternative that best completes the statement or answers the question. Solve the equation - x+7+5=x\sqrt{\mathrm{x}+7}+5=\mathrm{x}x+7​+5=x

Find the power of iii . - i3\mathrm{i}^{3}i3

Multiply, then simplify the product. Assume that all variables represent positive real numbers. - (3+3)2(3+\sqrt{3})^{2}(3+3​)2

Simplify the radical. Assume that all variables represent positive real numbers. - 276\sqrt[6]{27}627​

Solve the problem. Give the answer as a simplified radical expression. -Find the perimeter of a triangle with sides measuring 28,4502 \sqrt{8}, 4 \sqrt{50}28​,450​ , and 5805 \sqrt{80}580​ meters.

Explain how to evaluate (−64)4/3(-64)^{4 / 3}(−64)4/3 .

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

Write with radicals. Assume that all variables represent positive real numbers - $(5 x)^{3 / 7}+(4 y)^{3 / 5}$

Simplify the expression involving rational exponents. - $\left(\frac{64}{729}\right)^{-2 / 3}$

Choose the one alternative that best completes the statement or answers the question. Evaluate - $-\sqrt{196}$

Write the expression in the form a + bi. - $\frac{8 \mathrm{i}}{4+9 \mathrm{i}}$

Is $\sqrt{11}$ simplified?

Use the Pythagorean formula to find the exact length of side $b$ in the figure. -

Write with rational exponents and then apply the properties of exponents. Assume that all radicands represent positive real numbers. Give answers in exponential form. - $\sqrt[5]{\sqrt[4]{w}}$

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

Simplify the radical. Assume that all variables represent positive real numbers - $\sqrt[4]{\frac{625}{81}}$

Find the equation of a circle satisfying the given conditions. -Center: $(6,0)$ ; radius: 6

Simplify the expression. Assume that all variables represent positive real numbers. - $\frac{5^{3 / 4} x^{-2 / 5} y^{3 / 2}}{5^{-5 / 4} x^{13 / 5} y^{1 / 4}}$

Explain why if $\mathrm{n}$ is odd, $\mathrm{x}^{1 / \mathrm{n}}$ is defined even if $\mathrm{x}$ is negative.

Choose the one alternative that best completes the statement or answers the question. Solve the equation - $\sqrt{\mathrm{x}+7}+5=\mathrm{x}$

Find the power of $i$ . - $\mathrm{i}^{3}$

Multiply, then simplify the product. Assume that all variables represent positive real numbers. - $(3+\sqrt{3})^{2}$

Simplify the radical. Assume that all variables represent positive real numbers. - $\sqrt[6]{27}$

Solve the problem. Give the answer as a simplified radical expression. -Find the perimeter of a triangle with sides measuring $2 \sqrt{8}, 4 \sqrt{50}$ , and $5 \sqrt{80}$ meters.

Explain how to evaluate $(-64)^{4 / 3}$ .