Question 1

Simplify by factoring.
-$\sqrt{x^{13}}$

Accepted Answer

A)  $x^{6} \sqrt{x}$ 
B)  $x^{5} \sqrt{x}$ 
C)  $x^{6}$ 
D)  $x^{7}$ 
A)  $x^{6} \sqrt{x}$ 
B)  $x^{5} \sqrt{x}$ 
C)  $x^{6}$ 
D)  $x^{7}$

Question 2

Rationalize the denominator and simplify.
-$\frac{3}{8-\sqrt{2}}$

Accepted Answer

A)  $\frac{24-3 \sqrt{2}}{62}$ 
B)  $\frac{3}{8}-\frac{3}{\sqrt{2}}$ 
C)  $\frac{24+3 \sqrt{2}}{6}$ 
D)  $\frac{24+3 \sqrt{2}}{62}$ 
A)  $\frac{24-3 \sqrt{2}}{62}$ 
B)  $\frac{3}{8}-\frac{3}{\sqrt{2}}$ 
C)  $\frac{24+3 \sqrt{2}}{6}$ 
D)  $\frac{24+3 \sqrt{2}}{62}$

Question 3

Multiply and then, if possible, simplify by factoring.
-$\sqrt{\mathrm{rs}} \sqrt{\mathrm{st}}$

Accepted Answer

A)  $s \sqrt{\mathrm{rt}}$ 
B)  $\sqrt{\mathrm{rst}}$ 
C)  $s^{2} \sqrt{\mathrm{rt}}$ 
D)  $\sqrt{r s^{2} t}$ 
A)  $s \sqrt{\mathrm{rt}}$ 
B)  $\sqrt{\mathrm{rst}}$ 
C)  $s^{2} \sqrt{\mathrm{rt}}$ 
D)  $\sqrt{r s^{2} t}$

Question 4

Find the length of the third side of the right triangle. Assume that $c$ represents the length of the hypotenuse. Give an exact answer and, if appropriate, an approximation to three decimal places.
-$\mathrm{a}=1, \mathrm{c}=16$

Accepted Answer

A)  $b=\sqrt{257} ; b \approx 16.031$ 
B)  $b=\sqrt{15} ; b \approx 3.873$ 
C)  $b=15$ 
D)  $b=\sqrt{255} ; b \approx 15.969$ 
A)  $b=\sqrt{257} ; b \approx 16.031$ 
B)  $b=\sqrt{15} ; b \approx 3.873$ 
C)  $b=15$ 
D)  $b=\sqrt{255} ; b \approx 15.969$

Question 5

Simplify.
-$\sqrt{\frac{54 \mathrm{a}^{2}}{\mathrm{c}^{2}}}$

Accepted Answer

A)  $\frac{3 a \sqrt{6}}{c}$ 
B)  $\frac{3 \sqrt{6 a^{2}}}{\mathrm{c}}$ 
C)  $\frac{9 \mathrm{a} \sqrt{6}}{\mathrm{c}}$ 
D)  $\frac{\mathrm{a} \sqrt{54}}{\mathrm{C}}$ 
A)  $\frac{3 a \sqrt{6}}{c}$ 
B)  $\frac{3 \sqrt{6 a^{2}}}{\mathrm{c}}$ 
C)  $\frac{9 \mathrm{a} \sqrt{6}}{\mathrm{c}}$ 
D)  $\frac{\mathrm{a} \sqrt{54}}{\mathrm{C}}$

Question 6

Find all square roots of the number.
-441

Accepted Answer

A)  $21,-21$ 
B)  $23,-23$ 
C)  $22,-22$ 
D)  $42,-42$ 
A)  $21,-21$ 
B)  $23,-23$ 
C)  $22,-22$ 
D)  $42,-42$

Question 7

Rationalize the denominator and simplify.
-$\frac{\sqrt{\mathrm{x}^{5}}}{\sqrt{\mathrm{xy}^{8}}}$

Accepted Answer

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

Question 8

Solve the equation.
-$\sqrt{10+\mathrm{x}}=-1-\sqrt{11-5 \mathrm{x}}$

Accepted Answer

A)  1 
B)  No solution 
C)  0 
D)  $\frac{11}{2}$ 
A)  1 
B)  No solution 
C)  0 
D)  $\frac{11}{2}$

Question 9

Solve the equation.
-$\sqrt{2 \mathrm{k}+1}=7$

Accepted Answer

A)  12 
B)  24 
C)  48 
D)  -24 
A)  12 
B)  24 
C)  48 
D)  -24

Question 10

Solve the equation.
-$\sqrt{\mathrm{x}}=-3$

Accepted Answer

A)  9 
B)  6 
C)  No Solution 
D)  -9 
A)  9 
B)  6 
C)  No Solution 
D)  -9

Question 11

Simplify.
-$-\sqrt{16}$

Accepted Answer

A)  -4 
B)  Not a real number 
C)  -8 
D)  - 5 
A)  -4 
B)  Not a real number 
C)  -8 
D)  - 5

Question 12

Rationalize the denominator and simplify.
-$\frac{\sqrt{5}}{5 \sqrt{7}-\sqrt{5}}$

Accepted Answer

A)  $\frac{\sqrt{35}+1}{36}$ 
B)  $\frac{\sqrt{35}-1}{34}$ 
C)  $\frac{\sqrt{7}+1}{34}$ 
D)  $\frac{\sqrt{35}+1}{34}$ 
A)  $\frac{\sqrt{35}+1}{36}$ 
B)  $\frac{\sqrt{35}-1}{34}$ 
C)  $\frac{\sqrt{7}+1}{34}$ 
D)  $\frac{\sqrt{35}+1}{34}$

Question 13

Rationalize the denominator and simplify.
-$\sqrt{\frac{45 x^{3}}{y^{5}}}$

Accepted Answer

A)  $\frac{3 x \sqrt{5 x y}}{y^{3}}$ 
B)  $\frac{3 x^{3} \sqrt{5}}{y^{5}}$ 
C)  $\frac{x \sqrt{45 x y}}{y^{3}}$ 
D)  $\frac{3 x}{y^{2}} \sqrt{\frac{5 x}{y}}$ 
A)  $\frac{3 x \sqrt{5 x y}}{y^{3}}$ 
B)  $\frac{3 x^{3} \sqrt{5}}{y^{5}}$ 
C)  $\frac{x \sqrt{45 x y}}{y^{3}}$ 
D)  $\frac{3 x}{y^{2}} \sqrt{\frac{5 x}{y}}$

Question 14

Perform the indicated operation.
-$(\sqrt{8}+\sqrt{7})(\sqrt{8}-\sqrt{7})$

Accepted Answer

A)  15 
B)  $8-4 \sqrt{28}$ 
C)  $8-2 \sqrt{56}$ 
D)  1 
A)  15 
B)  $8-4 \sqrt{28}$ 
C)  $8-2 \sqrt{56}$ 
D)  1

Question 15

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

Accepted Answer

A)  $\frac{3 \sqrt{10}}{10}$ 
B)  $3 \sqrt{10}$ 
C)  103 
D)  $\frac{9 \sqrt{10}}{10}$ 
A)  $\frac{3 \sqrt{10}}{10}$ 
B)  $3 \sqrt{10}$ 
C)  103 
D)  $\frac{9 \sqrt{10}}{10}$

Question 16

Simplify by factoring.
-$2 \sqrt{21}$

Accepted Answer

A)  42 
B)  9 
C)  $21 \sqrt{2}$ 
D)  $2 \sqrt{21}$ 
A)  42 
B)  9 
C)  $21 \sqrt{2}$ 
D)  $2 \sqrt{21}$

Question 17

Solve the problem. Give an exact answer and an approximation to the nearest tenth.
-The diagram below shows the side view of a plan for a slanted roof. Find the unknown length in this roof plan.

Accepted Answer

A)  $\sqrt{274} \mathrm{ft} \approx 16.6 \mathrm{ft}$ 
B)  $\sqrt{22} \mathrm{ft} \approx 4.7 \mathrm{ft}$ 
C)  $\sqrt{274} \mathrm{ft}=137.0 \mathrm{ft}$ 
D)  $\sqrt{64} \mathrm{ft} \approx 8.0 \mathrm{ft}$ 
A)  $\sqrt{274} \mathrm{ft} \approx 16.6 \mathrm{ft}$ 
B)  $\sqrt{22} \mathrm{ft} \approx 4.7 \mathrm{ft}$ 
C)  $\sqrt{274} \mathrm{ft}=137.0 \mathrm{ft}$ 
D)  $\sqrt{64} \mathrm{ft} \approx 8.0 \mathrm{ft}$

Question 18

Find the length of the third side of the right triangle. Where appropriate, give both an exact answer and an approximation to three decimal places.
-

Accepted Answer

A)  20 
B)  19 
C)  14 
D)  16 
A)  20 
B)  19 
C)  14 
D)  16

Question 19

Rationalize the denominator and simplify.
-$\sqrt{\frac{9}{40}}$

Accepted Answer

A)  $3 \sqrt{10}$ 
B)  $\frac{3 \sqrt{10}}{20}$ 
C)  $\frac{3 \sqrt{10}}{10}$ 
D)  40 
A)  $3 \sqrt{10}$ 
B)  $\frac{3 \sqrt{10}}{20}$ 
C)  $\frac{3 \sqrt{10}}{10}$ 
D)  40

Question 20

Provide an appropriate response.
-Consider the following:
$\begin{gathered}
\sqrt{x}+3=5 \
\sqrt{x}=2 \
(\sqrt{x})^{2}=2^{2} \
x=4
\end{gathered}$
What would have happened if the radical had not been isolated before the principle of squaring was used? Could the equation still have been solved? Why or why not?

Accepted Answer

A radical term would have remained after

Simplify by factoring. - $\sqrt{x^{13}}$

Rationalize the denominator and simplify. - $\frac{3}{8-\sqrt{2}}$

Multiply and then, if possible, simplify by factoring. - $\sqrt{\mathrm{rs}} \sqrt{\mathrm{st}}$

Find the length of the third side of the right triangle. Assume that $c$ represents the length of the hypotenuse. Give an exact answer and, if appropriate, an approximation to three decimal places. - $\mathrm{a}=1, \mathrm{c}=16$

Simplify. - $\sqrt{\frac{54 \mathrm{a}^{2}}{\mathrm{c}^{2}}}$

Find all square roots of the number. -441

Rationalize the denominator and simplify. - $\frac{\sqrt{\mathrm{x}^{5}}}{\sqrt{\mathrm{xy}^{8}}}$

Solve the equation. - $\sqrt{10+\mathrm{x}}=-1-\sqrt{11-5 \mathrm{x}}$

Solve the equation. - $\sqrt{2 \mathrm{k}+1}=7$

Solve the equation. - $\sqrt{\mathrm{x}}=-3$

Simplify. - $-\sqrt{16}$

Rationalize the denominator and simplify. - $\frac{\sqrt{5}}{5 \sqrt{7}-\sqrt{5}}$

Rationalize the denominator and simplify. - $\sqrt{\frac{45 x^{3}}{y^{5}}}$

Perform the indicated operation. - $(\sqrt{8}+\sqrt{7})(\sqrt{8}-\sqrt{7})$

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

Simplify by factoring. - $2 \sqrt{21}$

Solve the problem. Give an exact answer and an approximation to the nearest tenth. -The diagram below shows the side view of a plan for a slanted roof. Find the unknown length in this roof plan.

Find the length of the third side of the right triangle. Where appropriate, give both an exact answer and an approximation to three decimal places. -

Rationalize the denominator and simplify. - $\sqrt{\frac{9}{40}}$

Provide an appropriate response. -Consider the following: +3=5 =2 (= x=4 What would have happened if the radical had not been isolated before the principle of squaring was used? Could the equation still have been solved? Why or why not?

Introduction to Real Numbers and Algebraic Expressions

Solving Equations and Inequalities

Graphs of Linear Equations

Polynomials: Operations

Polynomials: Factoring

Rational Expressions and Equations

Systems of Equations

Quadratic Equations

Equations of Lines and Simplifying Radical Expressions

Filters

Exam 8: Radical Expressions and Equations

Simplify by factoring. - x13\sqrt{x^{13}}x13​

Rationalize the denominator and simplify. - 38−2\frac{3}{8-\sqrt{2}}8−2​3​

Multiply and then, if possible, simplify by factoring. - rsst\sqrt{\mathrm{rs}} \sqrt{\mathrm{st}}rs​st​

Find the length of the third side of the right triangle. Assume that ccc represents the length of the hypotenuse. Give an exact answer and, if appropriate, an approximation to three decimal places. - a=1,c=16\mathrm{a}=1, \mathrm{c}=16a=1,c=16

Simplify. - 54a2c2\sqrt{\frac{54 \mathrm{a}^{2}}{\mathrm{c}^{2}}}c254a2​​