Question 1

Rationalize the denominator and simplify.
-$\frac{\sqrt{3}-10}{\sqrt{7}+4}$

Accepted Answer

A)  $\frac{\sqrt{21}+40-10 \sqrt{7}-4 \sqrt{3}}{-9}$ 
B)  $\frac{\sqrt{21}+40-10 \sqrt{7}-4 \sqrt{3}}{23}$ 
C)  $\frac{\sqrt{21}-40-10 \sqrt{7}+4 \sqrt{3}}{23}$ 
D)  $\frac{\sqrt{21}-40-10 \sqrt{7}+4 \sqrt{3}}{-9}$ 
A)  $\frac{\sqrt{21}+40-10 \sqrt{7}-4 \sqrt{3}}{-9}$ 
B)  $\frac{\sqrt{21}+40-10 \sqrt{7}-4 \sqrt{3}}{23}$ 
C)  $\frac{\sqrt{21}-40-10 \sqrt{7}+4 \sqrt{3}}{23}$ 
D)  $\frac{\sqrt{21}-40-10 \sqrt{7}+4 \sqrt{3}}{-9}$

Question 2

Multiply.
-$10 \mathrm{i} \cdot 2 \mathrm{i}$

Accepted Answer

A)  -20 
B)  - 12 
C)  20 
D)  12 
A)  -20 
B)  - 12 
C)  20 
D)  12

Question 3

Solve. Check for extraneous solutions.
-$\sqrt{41-2 x}+3=\sqrt{9-2 x}+7$

Accepted Answer

A)  $\left\{\frac{55}{2}\right\}$ 
B)  $\left\{\frac{7}{2}\right\}$ 
C)  $\left\{\frac{5}{2}\right\}$ 
D)  $\varnothing$ 
A)  $\left\{\frac{55}{2}\right\}$ 
B)  $\left\{\frac{7}{2}\right\}$ 
C)  $\left\{\frac{5}{2}\right\}$ 
D)  $\varnothing$

Question 4

How many real cube roots does any negative number have?

Accepted Answer

A)  None 
B)  Two 
C)  Three 
D)  One 
A)  None 
B)  Two 
C)  Three 
D)  One

Question 5

Multiply the conjugates.
-$(7-5 i)(7+5 i)$

Accepted Answer

A)  74 
B)  24 
C)  $49-25 \mathrm{i}$ 
D)  $49+25 \mathrm{i}$ 
A)  74 
B)  24 
C)  $49-25 \mathrm{i}$ 
D)  $49+25 \mathrm{i}$

Question 6

$7 \div(4 \mathrm{i})$

Accepted Answer

A)  $-\frac{7}{4} \mathrm{i}$ 
B)  $\frac{7}{4} \mathrm{i}$ 
C)  $\frac{4}{7} \mathrm{i}$ 
D)  $-\frac{4}{7} \mathrm{i}$ 
A)  $-\frac{7}{4} \mathrm{i}$ 
B)  $\frac{7}{4} \mathrm{i}$ 
C)  $\frac{4}{7} \mathrm{i}$ 
D)  $-\frac{4}{7} \mathrm{i}$

Question 7

Simplify.
-$\sqrt{2} \cdot \sqrt{50}$

Accepted Answer

A)  100 
B)  200 
C)  20 
D)  10 
A)  100 
B)  200 
C)  20 
D)  10

Question 8

Multiply.
-$-4 \mathrm{i} \cdot 8 \mathrm{i}$

Accepted Answer

A)  12 
B)  -12 
C)  -32 
D)  32 
A)  12 
B)  -12 
C)  -32 
D)  32

Question 9

$-\sqrt[4]{1024}$

Accepted Answer

A)  5 
B)  -5 
C)  $4 \sqrt[4]{4}$ 
D)  $-4 \sqrt[4]{4}$ 
A)  5 
B)  -5 
C)  $4 \sqrt[4]{4}$ 
D)  $-4 \sqrt[4]{4}$

Question 10

A ssume all variables represent nonnegative real numbers.
-$\sqrt{15 x^{3}} \cdot \sqrt{15 x^{5}}$

Accepted Answer

A)  $\sqrt{15 x^{4}}$ 
B)  $x^{4} \sqrt{30}$ 
C)  $15 x^{4}$ 
D)  $\sqrt{225 x^{8}}$ 
A)  $\sqrt{15 x^{4}}$ 
B)  $x^{4} \sqrt{30}$ 
C)  $15 x^{4}$ 
D)  $\sqrt{225 x^{8}}$

Question 11

Find the domain of the function. Express the answer in interval notation.
-$f(x)=\sqrt[3]{x-18}$

Accepted Answer

A)  $(-\infty, 18)$ 
B)  $[18, \infty)$ 
C)  $(-\infty,-18)$ 
D)  $(-\infty, \infty)$ 
A)  $(-\infty, 18)$ 
B)  $[18, \infty)$ 
C)  $(-\infty,-18)$ 
D)  $(-\infty, \infty)$

Question 12

Multiply the conjugates.
-$(-2-10 \mathrm{i})(-2+10 \mathrm{i})$

Accepted Answer

A)  -12 
B)  104 
C)  - 96 
D)  8 
A)  -12 
B)  104 
C)  - 96 
D)  8

Question 13

Explain what is wrong with this first step in the solution process for the equation $\sqrt{6 x+10}=\sqrt{-6 x-3} \cdot 6 x+10=6 x+3$.

Accepted Answer

The squaring of the

Question 14

A ssume all variables represent nonnegative real numbers.
-$\sqrt[3]{\mathrm{xy}^{5}} \cdot \sqrt[3]{\mathrm{x}^{16 \mathrm{y}^{8}}}$

Accepted Answer

A)  $x^{5} y^{4} \sqrt[3]{x y^{2}}$ 
B)  $x^{5} y^{6} \sqrt[3]{x^{2} y^{2}}$ 
C)  $x^{5} y^{4} \sqrt[3]{x^{2} y}$ 
D)  $x^{5} y^{5} \sqrt[3]{x y}$ 
A)  $x^{5} y^{4} \sqrt[3]{x y^{2}}$ 
B)  $x^{5} y^{6} \sqrt[3]{x^{2} y^{2}}$ 
C)  $x^{5} y^{4} \sqrt[3]{x^{2} y}$ 
D)  $x^{5} y^{5} \sqrt[3]{x y}$

Question 15

Simplify by rationalizing the denominator. Write your answer in a + bi form.
-$\frac{7+4 i}{8-4 i}$

Accepted Answer

A)  $18+1 \mathrm{i}$ 
B)  $\frac{3}{2}-\frac{1}{16} \mathrm{i}$ 
C)  $\frac{1}{2}+\frac{3}{4} \mathrm{i}$ 
D)  $\frac{1}{24}-\frac{1}{16} \mathrm{i}$ 
A)  $18+1 \mathrm{i}$ 
B)  $\frac{3}{2}-\frac{1}{16} \mathrm{i}$ 
C)  $\frac{1}{2}+\frac{3}{4} \mathrm{i}$ 
D)  $\frac{1}{24}-\frac{1}{16} \mathrm{i}$

Question 16

Multiply
-$(5 \sqrt{7}-4 \sqrt{11})^{2}$

Accepted Answer

A)  $-9-40 \sqrt{77}$ 
B)  351 
C)  $351-40 \sqrt{77}$ 
D)  $351-20 \sqrt{77}$ 
A)  $-9-40 \sqrt{77}$ 
B)  351 
C)  $351-40 \sqrt{77}$ 
D)  $351-20 \sqrt{77}$

Question 17

Solve. Check for extraneous solutions.
-$\sqrt{\mathrm{x}^{2}+2 \mathrm{x}+81}=9$

Accepted Answer

A)  $\{0,-2\}$ 
B)  $\{-2,2\}$ 
C)  $\{-2\}$ 
D)  $\varnothing$ 
A)  $\{0,-2\}$ 
B)  $\{-2,2\}$ 
C)  $\{-2\}$ 
D)  $\varnothing$

Question 18

What is the simplest radical expression that can be multiplied by the given radical expression so that the product can be simplified to no longer contain a radical?
-$\sqrt[3]{225}$

Accepted Answer

A)  $\sqrt[3]{30}$ 
B)  $\sqrt[3]{9}$ 
C)  $\sqrt[3]{15}$ 
D)  $\sqrt[3]{5}$ 
A)  $\sqrt[3]{30}$ 
B)  $\sqrt[3]{9}$ 
C)  $\sqrt[3]{15}$ 
D)  $\sqrt[3]{5}$

Question 19

$(5+5 \mathrm{i}) \div(8-7 \mathrm{i})$

Accepted Answer

A)  $\frac{75}{113}-\frac{5}{113} \mathrm{i}$ 
B)  $\frac{1}{3}$ 
C)  $\frac{5}{113}+\frac{75}{113} \mathrm{i}$ 
D)  5 - $5 \mathrm{i}$ 
A)  $\frac{75}{113}-\frac{5}{113} \mathrm{i}$ 
B)  $\frac{1}{3}$ 
C)  $\frac{5}{113}+\frac{75}{113} \mathrm{i}$ 
D)  5 - $5 \mathrm{i}$

Question 20

Simplify the expression. Assume all variables represent nonnegative real numbers.
-$\frac{\sqrt[4]{y^{5}}}{\sqrt[7]{y^{5}}}$

Accepted Answer

A)  $\frac{\sqrt[4]{y^{5}}}{\sqrt[7]{y^{5}}}$ 
B)  $\sqrt[3]{y^{5}}$ 
C)  $\sqrt[11]{y^{5}}$ 
D)  $\sqrt[28]{y^{15}}$ 
A)  $\frac{\sqrt[4]{y^{5}}}{\sqrt[7]{y^{5}}}$ 
B)  $\sqrt[3]{y^{5}}$ 
C)  $\sqrt[11]{y^{5}}$ 
D)  $\sqrt[28]{y^{15}}$

Rationalize the denominator and simplify. - $\frac{\sqrt{3}-10}{\sqrt{7}+4}$

Multiply. - $10 \mathrm{i} \cdot 2 \mathrm{i}$

Solve. Check for extraneous solutions. - $\sqrt{41-2 x}+3=\sqrt{9-2 x}+7$

How many real cube roots does any negative number have?

Multiply the conjugates. - $(7-5 i)(7+5 i)$

$7 \div(4 \mathrm{i})$

Simplify. - $\sqrt{2} \cdot \sqrt{50}$

Multiply. - $-4 \mathrm{i} \cdot 8 \mathrm{i}$

$-\sqrt[4]{1024}$

A ssume all variables represent nonnegative real numbers. - $\sqrt{15 x^{3}} \cdot \sqrt{15 x^{5}}$

Find the domain of the function. Express the answer in interval notation. - $f(x)=\sqrt[3]{x-18}$

Multiply the conjugates. - $(-2-10 \mathrm{i})(-2+10 \mathrm{i})$

Explain what is wrong with this first step in the solution process for the equation $\sqrt{6 x+10}=\sqrt{-6 x-3} \cdot 6 x+10=6 x+3$ .

A ssume all variables represent nonnegative real numbers. - $\sqrt[3]{\mathrm{xy}^{5}} \cdot \sqrt[3]{\mathrm{x}^{16 \mathrm{y}^{8}}}$

Simplify by rationalizing the denominator. Write your answer in a + bi form. - $\frac{7+4 i}{8-4 i}$

Multiply - $(5 \sqrt{7}-4 \sqrt{11})^{2}$

Solve. Check for extraneous solutions. - $\sqrt{\mathrm{x}^{2}+2 \mathrm{x}+81}=9$

What is the simplest radical expression that can be multiplied by the given radical expression so that the product can be simplified to no longer contain a radical? - $\sqrt[3]{225}$

$(5+5 \mathrm{i}) \div(8-7 \mathrm{i})$

Simplify the expression. Assume all variables represent nonnegative real numbers. - $\frac{\sqrt[4]{y^{5}}}{\sqrt[7]{y^{5}}}$

Review of Real Numbers

Linear and Absolute Value Equations and Inequalities

Graphing Linear Equations

Systems of Equations

Exponents, Polynomials, and Factoring Polynomials

Rational Expressions and Equations

Quadratic Equations

Logarithmic and Exponential Functions

Conic Sections

Sequences, Series, and the Binomial Theorem

Filters

Exam 7: Radical Expressions and Equations

Rationalize the denominator and simplify. - 3−107+4\frac{\sqrt{3}-10}{\sqrt{7}+4}7​+43​−10​

Multiply. - 10i⋅2i10 \mathrm{i} \cdot 2 \mathrm{i}10i⋅2i

Solve. Check for extraneous solutions. - 41−2x+3=9−2x+7\sqrt{41-2 x}+3=\sqrt{9-2 x}+741−2x​+3=9−2x​+7

How many real cube roots does any negative number have?

Multiply the conjugates. - (7−5i)(7+5i)(7-5 i)(7+5 i)(7−5i)(7+5i)

7÷(4i)7 \div(4 \mathrm{i})7÷(4i)

Simplify. - 2⋅50\sqrt{2} \cdot \sqrt{50}2​⋅50​

Multiply. - −4i⋅8i-4 \mathrm{i} \cdot 8 \mathrm{i}−4i⋅8i

−10244-\sqrt[4]{1024}−41024​

A ssume all variables represent nonnegative real numbers. - 15x3⋅15x5\sqrt{15 x^{3}} \cdot \sqrt{15 x^{5}}15x3​⋅15x5​

Find the domain of the function. Express the answer in interval notation. - f(x)=x−183f(x)=\sqrt[3]{x-18}f(x)=3x−18​

Multiply the conjugates. - (−2−10i)(−2+10i)(-2-10 \mathrm{i})(-2+10 \mathrm{i})(−2−10i)(−2+10i)

Explain what is wrong with this first step in the solution process for the equation 6x+10=−6x−3⋅6x+10=6x+3\sqrt{6 x+10}=\sqrt{-6 x-3} \cdot 6 x+10=6 x+36x+10​=−6x−3​⋅6x+10=6x+3 .

A ssume all variables represent nonnegative real numbers. - xy53⋅x16y83\sqrt[3]{\mathrm{xy}^{5}} \cdot \sqrt[3]{\mathrm{x}^{16 \mathrm{y}^{8}}}3xy5​⋅3x16y8​

Simplify by rationalizing the denominator. Write your answer in a + bi form. - 7+4i8−4i\frac{7+4 i}{8-4 i}8−4i7+4i​

Multiply - (57−411)2(5 \sqrt{7}-4 \sqrt{11})^{2}(57​−411​)2

Solve. Check for extraneous solutions. - x2+2x+81=9\sqrt{\mathrm{x}^{2}+2 \mathrm{x}+81}=9x2+2x+81​=9

What is the simplest radical expression that can be multiplied by the given radical expression so that the product can be simplified to no longer contain a radical? - 2253\sqrt[3]{225}3225​

(5+5i)÷(8−7i)(5+5 \mathrm{i}) \div(8-7 \mathrm{i})(5+5i)÷(8−7i)

Simplify the expression. Assume all variables represent nonnegative real numbers. - y54y57\frac{\sqrt[4]{y^{5}}}{\sqrt[7]{y^{5}}}7y5​4y5​​

Review of Real Numbers

Linear and Absolute Value Equations and Inequalities

Graphing Linear Equations

Systems of Equations

Exponents, Polynomials, and Factoring Polynomials

Rational Expressions and Equations

Quadratic Equations

Logarithmic and Exponential Functions

Conic Sections

Sequences, Series, and the Binomial Theorem

Filters

Rationalize the denominator and simplify. - $\frac{\sqrt{3}-10}{\sqrt{7}+4}$

Multiply. - $10 \mathrm{i} \cdot 2 \mathrm{i}$

Solve. Check for extraneous solutions. - $\sqrt{41-2 x}+3=\sqrt{9-2 x}+7$

Multiply the conjugates. - $(7-5 i)(7+5 i)$

$7 \div(4 \mathrm{i})$

Simplify. - $\sqrt{2} \cdot \sqrt{50}$

Multiply. - $-4 \mathrm{i} \cdot 8 \mathrm{i}$

$-\sqrt[4]{1024}$

A ssume all variables represent nonnegative real numbers. - $\sqrt{15 x^{3}} \cdot \sqrt{15 x^{5}}$

Find the domain of the function. Express the answer in interval notation. - $f(x)=\sqrt[3]{x-18}$

Multiply the conjugates. - $(-2-10 \mathrm{i})(-2+10 \mathrm{i})$

Explain what is wrong with this first step in the solution process for the equation $\sqrt{6 x+10}=\sqrt{-6 x-3} \cdot 6 x+10=6 x+3$ .

A ssume all variables represent nonnegative real numbers. - $\sqrt[3]{\mathrm{xy}^{5}} \cdot \sqrt[3]{\mathrm{x}^{16 \mathrm{y}^{8}}}$

Simplify by rationalizing the denominator. Write your answer in a + bi form. - $\frac{7+4 i}{8-4 i}$

Multiply - $(5 \sqrt{7}-4 \sqrt{11})^{2}$

Solve. Check for extraneous solutions. - $\sqrt{\mathrm{x}^{2}+2 \mathrm{x}+81}=9$

What is the simplest radical expression that can be multiplied by the given radical expression so that the product can be simplified to no longer contain a radical? - $\sqrt[3]{225}$

$(5+5 \mathrm{i}) \div(8-7 \mathrm{i})$

Simplify the expression. Assume all variables represent nonnegative real numbers. - $\frac{\sqrt[4]{y^{5}}}{\sqrt[7]{y^{5}}}$