Question 1

Simplify. Assume that all variables represent positive real numbers.
-$\sqrt[3]{64 \mathrm{a}^{8} \mathrm{~b}^{5}}$
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) $4 a b \sqrt[3]{a^{3} b^{3}}$

Accepted Answer

The answer of Simplify. Assume that all variables represent positive...

Question 2

Simplify. Assume that all variables represent positive real numbers.
-$\sqrt{3} \cdot \sqrt[3]{4}$
A) $\sqrt[6]{108}$
B) $\sqrt[6]{432}$
C) $\sqrt[6]{144}$
D) $\sqrt[6]{12}$

Accepted Answer

The answer of Simplify. Assume that all variables represent positive...

Question 3

Simplify. Assume that all variables represent positive real numbers.
-$\sqrt{192}-3 \sqrt{48}-3 \sqrt{108}$
A) $36 \sqrt{3}$
B) $-22 \sqrt{3}$
C) $-48 \sqrt{3}$
D) $22 \sqrt{3}$

Accepted Answer

First, simplify each term under the square root by factoring out perfect squares: $\sqrt{192} = \sqrt{64*3} = 8\sqrt{3}$, $\sqrt{48} = \sqrt{16*3} = 4\sqrt{3}$, and $\sqrt{108} = \sqrt{36*3} = 6\sqrt{3}$. Then, combine like terms: $8\sqrt{3} - 3(4\sqrt{3}) - 3(6\sqrt{3}) = 8\sqrt{3} - 12\sqrt{3} - 18\sqrt{3} = -22\sqrt{3}$.

Question 4

Simplify. Assume that all variables represent positive real numbers.
-$\sqrt[3]{32 x^{9} y^{7}}-\sqrt[3]{108 x^{12} y^{4}}$

A) $\left(2 x^{3} y^{2}-3 x^{4} y\right) \sqrt[3]{4 y}$
B) $\left(2 x^{3} y^{2}+3 x^{4} y\right) \sqrt[3]{4 y}$
C) $\left(2 x^{3} y^{2}+3 x^{4} y\right) \sqrt[3]{4}$
D) $\left(2 x^{6} y^{4}-3 x^{9} y\right) \sqrt[3]{4}$

Accepted Answer

First, simplify each term inside the cube roots: $\sqrt[3]{32 x^{9} y^{7}} = 2x^3y^{\frac{7}{3}}$ and $\sqrt[3]{108 x^{12} y^{4}} = 3x^4y^{\frac{4}{3}}$. Factoring out the common term $\sqrt[3]{4y}$ gives $(2 x^{3} y^{2}-3 x^{4} y) \sqrt[3]{4 y}$.

Question 5

Simplify. Assume that all variables represent positive real numbers.
-$(3 \sqrt{3}+8)(5 \sqrt{3}-5)$
A) $85+25 \sqrt{3}$
B) $5+55 \sqrt{3}$
C) $5+25 \sqrt{3}$
D) $30 \sqrt{3}$

Accepted Answer

The answer of Simplify. Assume that all variables represent positive...

Question 6

Simplify. Assume that all variables represent positive real numbers.
-$(5-2 \sqrt{5})^{2}$
A) $45+20 \sqrt{5}$
B) $25-4 \sqrt{5}$
C) $25+4 \sqrt{5}$
D) $45-20 \sqrt{5}$

Accepted Answer

Expanding $(5-2 \sqrt{5})^{2}$ gives $5^2 - 2 \cdot 5 \cdot 2\sqrt{5} + (2\sqrt{5})^2 = 25 - 20\sqrt{5} + 20 = 45 - 20\sqrt{5}$.

Question 7

Simplify. Assume that all variables represent positive real numbers.
-$\frac{-3}{\sqrt{15}}$
A) $-\frac{\sqrt{15}}{5}$
B) $\frac{\sqrt{15}}{5}$
C) -3
D) $-\sqrt{3}$

Accepted Answer

To simplify $\frac{-3}{\sqrt{15}}$, you can rationalize the denominator by multiplying both the numerator and the denominator by $\sqrt{15}$. This gives $\frac{-3\sqrt{15}}{15}$, which simplifies to $-\frac{\sqrt{15}}{5}$.

Question 8

Simplify. Assume that all variables represent positive real numbers.
-$\frac{-50}{\sqrt{13}+\sqrt{3}}$
A) $-(\sqrt{13}-\sqrt{3})$
B) $-5(\sqrt{13}-\sqrt{3})$
C) $5(\sqrt{13}-\sqrt{3})$
D) $-5(\sqrt{13}+\sqrt{3})$

Accepted Answer

To simplify the expression $\frac{-50}{\sqrt{13}+\sqrt{3}}$, we multiply the numerator and the denominator by the conjugate of the denominator, which is $\sqrt{13}-\sqrt{3}$. This process eliminates the square root in the denominator. The simplification leads to option B, $-5(\sqrt{13}-\sqrt{3})$, as the correct answer.

Question 9

Write the fraction in lowest terms.
-$\frac{15+\sqrt{45}}{3}$
A) $5+\sqrt{5}$
B) $5+\sqrt{15}$
C) $15+\sqrt{5}$
D) $5+3 \sqrt{5}$

Accepted Answer

First, simplify the numerator: $15 + \sqrt{45} = 15 + \sqrt{9 \cdot 5} = 15 + 3\sqrt{5}$. Then, divide the entire expression by 3: $\frac{15}{3} + \frac{3\sqrt{5}}{3} = 5 + \sqrt{5}$, which matches option A.

Question 10

The length a spring is stretched from its natural length with work, W foot-pounds, is given by
$$\mathrm{L}=\sqrt{\frac{2 \mathrm{~W}}{\mathrm{k}}}$$
Where $\mathrm{k}$ is a constant for the given spring. If a certain spring has a constant of 74.8, and the spring is to be stretched 1.9 feet from its natural length, how much work will be necessary? Round your answer to the nearest tenth.
A) 51.6 foot-pounds
B) 135 foot-pounds
C) 270 foot-pounds
D) 71.1 foot-pounds

Accepted Answer

The answer of The length a spring is stretched from...

Question 11

Solve the equation.
-$\sqrt[3]{10 x}=\sqrt[3]{7 x-9}$
A) $\{-27\}$
B) $\{-3\}$
C) $\varnothing$
D) $\{3\}$

Accepted Answer

By cubing both sides of the equation $\sqrt[3]{10x} = \sqrt[3]{7x - 9}$, we get $10x = 7x - 9$. Solving for $x$, we find $x = -3$.

Question 12

Solve the equation.
-$5 x+\sqrt{x+56}=8+3 x$
A) $\{64\}$
B) $\{16\}$
C) $\left\{8, \frac{1}{4}\right\}$
D) $\left\{-8,-\frac{1}{4}\right\}$

Accepted Answer

The answer of Solve the equation.
-$5 x+\sqrt{x+56}=8+3 x$
A) $\{64\}$
B) $\{16\}$
C)...

Question 13

Solve the equation.
-$\sqrt{x+6}+\sqrt{2-x}=4$
A) $\{0\}$
B) $\{2,-2\}$
C) $\{-2\}$
D) $\{\sqrt{31},-2\}$

Accepted Answer

To solve $\sqrt{x+6}+\sqrt{2-x}=4$, we test the given options. For $x = -2$, $\sqrt{-2+6} + \sqrt{2-(-2)} = \sqrt{4} + \sqrt{4} = 2 + 2 = 4$, which satisfies the equation. For other options, the equation does not hold.

Question 14

Perform the indicated operation. Give answer in standard form.
-$(-3+4 \mathrm{i})-(6+5 \mathrm{i})-14 \mathrm{i}$
A) $9-15 \mathrm{i}$
B) $-9-5 \mathrm{i}$
C) $-9+15 i$
D) $-9-15 \mathrm{i}$

Accepted Answer

To solve, subtract the real parts and the imaginary parts separately: $(-3 - 6) + (4i - 5i - 14i) = -9 - 15i$.

Question 15

Perform the indicated operation. Give answer in standard form.
-$(6+8 \mathrm{i})(9+7 \mathrm{i})$
A) $56 \mathrm{i}^{2}+114 \mathrm{i}+54$
B) $-2-114 \mathrm{i}$
C) $-2+114 i$
D) $110+30 \mathrm{i}$

Accepted Answer

The answer of Perform the indicated operation. Give answer in...

Question 16

Perform the indicated operation. Give answer in standard form.
-$\frac{11+i}{1-i}$
A) $6+6 \mathrm{i}$
B) $5+6 \mathrm{i}$
C) $5+12 \mathrm{i}$
D) $5+5 \mathrm{i}$

Accepted Answer

The answer of Perform the indicated operation. Give answer in...

Question 17

Perform the indicated operation. Give answer in standard form.
-$\mathrm{i}^{5}$
A) -1
B) $-\mathrm{i}$
C) $\mathrm{i}$
D) 1

Accepted Answer

$\mathrm{i}^{5} = \mathrm{i}^{4} \cdot \mathrm{i} = 1 \cdot \mathrm{i} = \mathrm{i}$, because $\mathrm{i}^{4} = 1$.

Question 18

$\mathrm{i}^{3}=-\mathrm{i}$

Accepted Answer

$\mathrm{i}^{3} = \mathrm{i}^{2} \cdot \mathrm{i} = -1 \cdot \mathrm{i} = -\mathrm{i}$.

Quiz 8: Roots, Radicals, and Root Functions

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

Simplify. Assume that all variables represent positive real numbers. - $\sqrt{3} \cdot \sqrt[3]{4}$

Simplify. Assume that all variables represent positive real numbers. - $\sqrt{192}-3 \sqrt{48}-3 \sqrt{108}$

Simplify. Assume that all variables represent positive real numbers. - $\sqrt[3]{32 x^{9} y^{7}}-\sqrt[3]{108 x^{12} y^{4}}$

Simplify. Assume that all variables represent positive real numbers. - $(3 \sqrt{3}+8) (5 \sqrt{3}-5)$

Simplify. Assume that all variables represent positive real numbers. - $(5-2 \sqrt{5}) ^{2}$

Simplify. Assume that all variables represent positive real numbers. - $\frac{-3}{\sqrt{15}}$

Simplify. Assume that all variables represent positive real numbers. - $\frac{-50}{\sqrt{13}+\sqrt{3}}$

Write the fraction in lowest terms. - $\frac{15+\sqrt{45}}{3}$

Solve the equation. - $\sqrt[3]{10 x}=\sqrt[3]{7 x-9}$

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

Solve the equation. - $\sqrt{x+6}+\sqrt{2-x}=4$

Perform the indicated operation. Give answer in standard form. - $(-3+4 \mathrm{i}) -(6+5 \mathrm{i}) -14 \mathrm{i}$

Perform the indicated operation. Give answer in standard form. - $(6+8 \mathrm{i}) (9+7 \mathrm{i})$

Perform the indicated operation. Give answer in standard form. - $\frac{11+i}{1-i}$

Perform the indicated operation. Give answer in standard form. - $\mathrm{i}^{5}$

$\mathrm{i}^{3}=-\mathrm{i}$

Quiz 8: Roots, Radicals, and Root Functions

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

Simplify. Assume that all variables represent positive real numbers. - 3⋅43\sqrt{3} \cdot \sqrt[3]{4}3​⋅34​

Simplify. Assume that all variables represent positive real numbers. - 192−348−3108\sqrt{192}-3 \sqrt{48}-3 \sqrt{108}192​−348​−3108​

Simplify. Assume that all variables represent positive real numbers. - 32x9y73−108x12y43\sqrt[3]{32 x^{9} y^{7}}-\sqrt[3]{108 x^{12} y^{4}}332x9y7​−3108x12y4​

Simplify. Assume that all variables represent positive real numbers. - (33+8) (53−5) (3 \sqrt{3}+8) (5 \sqrt{3}-5) (33​+8) (53​−5)

Simplify. Assume that all variables represent positive real numbers. - (5−25) 2(5-2 \sqrt{5}) ^{2}(5−25​) 2

Simplify. Assume that all variables represent positive real numbers. - −315\frac{-3}{\sqrt{15}}15​−3​

Simplify. Assume that all variables represent positive real numbers. - −5013+3\frac{-50}{\sqrt{13}+\sqrt{3}}13​+3​−50​

Write the fraction in lowest terms. - 15+453\frac{15+\sqrt{45}}{3}315+45​​

Solve the equation. - 10x3=7x−93\sqrt[3]{10 x}=\sqrt[3]{7 x-9}310x​=37x−9​

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

Solve the equation. - x+6+2−x=4\sqrt{x+6}+\sqrt{2-x}=4x+6​+2−x​=4

Perform the indicated operation. Give answer in standard form. - (−3+4i) −(6+5i) −14i(-3+4 \mathrm{i}) -(6+5 \mathrm{i}) -14 \mathrm{i}(−3+4i) −(6+5i) −14i

Perform the indicated operation. Give answer in standard form. - (6+8i) (9+7i) (6+8 \mathrm{i}) (9+7 \mathrm{i}) (6+8i) (9+7i)

Perform the indicated operation. Give answer in standard form. - 11+i1−i\frac{11+i}{1-i}1−i11+i​

Perform the indicated operation. Give answer in standard form. - i5\mathrm{i}^{5}i5

i3=−i\mathrm{i}^{3}=-\mathrm{i}i3=−i

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

Simplify. Assume that all variables represent positive real numbers. - $\sqrt{3} \cdot \sqrt[3]{4}$

Simplify. Assume that all variables represent positive real numbers. - $\sqrt{192}-3 \sqrt{48}-3 \sqrt{108}$

Simplify. Assume that all variables represent positive real numbers. - $\sqrt[3]{32 x^{9} y^{7}}-\sqrt[3]{108 x^{12} y^{4}}$

Simplify. Assume that all variables represent positive real numbers. - $(3 \sqrt{3}+8) (5 \sqrt{3}-5)$

Simplify. Assume that all variables represent positive real numbers. - $(5-2 \sqrt{5}) ^{2}$

Simplify. Assume that all variables represent positive real numbers. - $\frac{-3}{\sqrt{15}}$

Simplify. Assume that all variables represent positive real numbers. - $\frac{-50}{\sqrt{13}+\sqrt{3}}$

Write the fraction in lowest terms. - $\frac{15+\sqrt{45}}{3}$

Solve the equation. - $\sqrt[3]{10 x}=\sqrt[3]{7 x-9}$

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

Solve the equation. - $\sqrt{x+6}+\sqrt{2-x}=4$

Perform the indicated operation. Give answer in standard form. - $(-3+4 \mathrm{i}) -(6+5 \mathrm{i}) -14 \mathrm{i}$

Perform the indicated operation. Give answer in standard form. - $(6+8 \mathrm{i}) (9+7 \mathrm{i})$

Perform the indicated operation. Give answer in standard form. - $\frac{11+i}{1-i}$

Perform the indicated operation. Give answer in standard form. - $\mathrm{i}^{5}$

$\mathrm{i}^{3}=-\mathrm{i}$