Deck 7: Radical Expressions and Equations

Question 1

Simplify the radical expression. Indicate if the expression is not a real number.

- $\sqrt{\frac{81}{16}}$

A) $\frac{\sqrt{9}}{4}$

B) $\frac{\sqrt{9}}{\sqrt{4}}$

C) $\frac{9}{4}$

D) 2

Answer 1

$\frac{9}{4}$

Question 2

Simplify the radical expression. Indicate if the expression is not a real number.

- $-\sqrt{361}$

A) -19
B) Not a real number
C) -180
D) 19

Answer 2

-19

Question 3

Simplify the radical expression. Indicate if the expression is not a real number.

- $-\sqrt{\frac{625}{36}}$

A) $\frac{25}{6}$

B) $-\frac{25}{6}$

C) Not a real number

D) $-\frac{312}{18}$

Answer 3

$-\frac{25}{6}$

Question 4

Simplify the radical expression. Indicate if the expression is not a real number.

- $\sqrt{\frac{484}{25}}$

A) $\frac{22}{6}$

B) $\frac{23}{5}$

C) $\frac{22}{5}$

D) $\frac{242}{12}$

Question 5

Simplify the radical expression. Indicate if the expression is not a real number.

- $\sqrt{-361}$

A) 19
B) -19
C) Not a real number
D) 180

Question 6

Simplify the radical expression. Indicate if the expression is not a real number.

- $\sqrt{-\frac{361}{36}}$

A) $\frac{19}{6}$
B) Not a real number
C) - 10
D) $-\frac{19}{6}$

Question 7

Simplify the radical expression. If appropriate, include absolute values.

- $-\sqrt[4]{x^{28}}$

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

Question 8

Simplify the radical expression. If appropriate, include absolute values.

- $\sqrt{144 x^{2}}$

A) $144 \mathrm{x}$
B) - 12|x|
C) $12 x$
D) $12|x|$

Question 9

Simplify the radical expression. If appropriate, include absolute values.

- $\sqrt{196 y^{6}}$

A) $14 y^{3}$
B) $14|\mathrm{y}|$
C) $14 y^{2}$
D) $14\left|y^{3}\right|$

Question 10

Simplify the radical expression. If appropriate, include absolute values.

- $\sqrt{(b+6)^{2}}$

A) $|b+6|$
B) $b+36$
C) $|b-6|$
D) $\sqrt{b}+6$

Question 11

Simplify the radical expression. If appropriate, include absolute values.

- $\sqrt{4 x^{2}+24 x+36}$

A) $|2 x+6|$
B) $2 x+6+\sqrt{24 x}$
C) $|2 x-6|$
D) $-2 x+8$

Question 12

Simplify the radical expression. If appropriate, include absolute values.

- $\sqrt{z^{2}-20 z+100}$

A) $-z+10$
B) $z-10 \mid$
C) $\mathrm{z}-10$
D) $|z|-10$

Question 13

Simplify the radical expression. If appropriate, include absolute values.

- $\sqrt{25 x^{2}+40 x+16}$

A) $|5 x+4|$
B) $|5 x|+4$
C) $5 x+4$
D) $5 x+\sqrt{20 x}$

Question 14

Find the missing number

- $\sqrt{?}=17$

A) 19
B) 34
C) 578
D) 289

Question 15

Find the missing number

- $\sqrt{?}=5 x$

A) $-5 x$
B) $25 x^{2}$
C) $5 x^{2}$
D) $25 x$

Question 16

Find the missing number

- $\sqrt{?}=12 \mathrm{y}^{3}$

A) $12 y^{3}$
B) $12 y^{6}$
C) $144 y^{6}$
D) $144 y^{3}$

Question 17

Find the missing number

- $\sqrt{20} \cdot \sqrt{?}=10$

A) $\frac{5}{2}$
B) 10
C) $\frac{1}{2}$
D) 5

Question 18

Find the missing number

- $\frac{\sqrt{?}}{\sqrt{5}}=8$

A) $8 \sqrt{5}$
B) 40
C) $\frac{8}{5}$
D) 320

Question 19

Approximate to the nearest thousandth, using a calculator

- $\sqrt{7}$

A) 2.646
B) 2.643
C) 2.651
D) 7.000

Question 20

Approximate to the nearest thousandth, using a calculator

- $-\sqrt{77}$

A) -8.772
B) -8.780
C) -77.000
D) -8.775

Question 21

Approximate to the nearest thousandth, using a calculator

- $\sqrt{417}$

A) 417.000
B) 20.426
C) 20.421
D) 20.418

Question 22

Approximate to the nearest thousandth, using a calculator

- $-\sqrt{9791}$

A) -98.949
B) -98.946
C) -98.954
D) -9791.000

Question 23

Approximate to the nearest thousandth, using a calculator

- $\sqrt{1.09}$

A) 1.000
B) 1.044
C) 1.059
D) 1.031

Question 24

Approximate to the nearest thousandth, using a calculator

- $\sqrt{0.000113}$

A) 0.011
B) 0.001
C) 1.100
D) 0.110

Question 25

Approximate to the nearest thousandth, using a calculator

- $\sqrt{0.00227}$

A) 4.800
B) 0.048
C) 0.480
D) 0.005

Question 26

Use the Babylonian method to approximate the given square root to the nearest thousandth.

- $\sqrt{47}$

A) 6.928
B) 6.782
C) 6.856
D) 7.071

Question 27

Use the Babylonian method to approximate the given square root to the nearest thousandth.

- $\sqrt{600}$

A) 24.454
B) 24.474
C) 24.495
D) 24.597

Question 28

Simplify the radical expression. Assume all variables represent nonnegative real numbers.

- $\sqrt[3]{216}$

A) 15
B) -6
C) 36
D) 6

Question 29

Simplify the radical expression. Assume all variables represent nonnegative real numbers.

- $-\sqrt[3]{64}$

A) 16
B) -4
C) 4
D) - 64

Question 30

Simplify the radical expression. Assume all variables represent nonnegative real numbers.

- $-\sqrt[3]{-512}$

A) -64
B) 23
C) -8
D) 8

Question 31

Simplify the radical expression. Assume all variables represent nonnegative real numbers.

- $\sqrt[4]{256}$

A) 3
B) 4
C) 16
D) 5

Question 32

Simplify the radical expression. Assume all variables represent nonnegative real numbers.

- $-\sqrt[3]{-64 \mathrm{x}^{3}}$

A) $-4 x$
B) $4 x$
C) $4 x^{3}$
D) $-64 x$

Question 33

Simplify the radical expression. Assume all variables represent nonnegative real numbers.

- $\sqrt[4]{81 x^{8}}$

A) 324
B) $3 x^{2}$
C) $-3 x^{2}$
D) $9 x^{2}$

Question 34

Simplify.

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

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

Question 35

Simplify.

- $\sqrt{8} \cdot \sqrt{2}$

A) $2 \sqrt{4}$
B) 16
C) 4
D) $2 \sqrt{2}$

Question 36

Simplify.

- $\frac{\sqrt{729}}{\sqrt{9}}$

A) 3
B) 81
C) 9
D) 27

Question 37

Simplify.

- $\frac{\sqrt{10}}{\sqrt{2}}$

A) $\sqrt{2}$
B) 5
C) $\sqrt{5}$
D) 2

Question 38

Simplify.

- $\sqrt[3]{49} \cdot \sqrt[3]{7}$

A) $\sqrt[3]{343}$
B) 7
C) 18
D) $7 \sqrt[3]{7}$

Question 39

Simplify.

- $\frac{\sqrt[3]{640}}{\sqrt[3]{10}}$

A) 3
B) 4
C) 5
D) 6

Question 40

Evaluate the radical function at the given value of the variable.

- $f(x)=\sqrt{x+9}+\sqrt{x}$ ; find $f(16)$ .

A) 7
B) 3
C) 9
D) 6

Question 41

Evaluate the radical function at the given value of the variable.

- $f(x)=-\sqrt{x^{2}+2 x+1}$ ; find $f(-9)$ .

A) 8
B) -8
C) 9
D) -9

Question 42

Evaluate the radical function at the given value of the variable.

- $f(x)=\sqrt{x^{2}+2 x+1}$ ; find $f(7)$ .

A) 7
B) 6
C) 8
D) 5

Question 43

The rms velocity (or average velocity) of a gas particle in meters per second is given by the formula $\mu_{\mathrm{rms}}=\sqrt{\frac{24.9435 \mathrm{~T}}{\mathrm{M}}}$ , where $\mathrm{T}$ is the absolute temperature in kelvin $\left(\mathrm{T}=\mathrm{T}_{\mathrm{C}}+273\right)$ and $\mathrm{M}$ is the molar mass (mass of 1 mole) of the gas in kilograms. Find the rms velocity for a molecule of Hydrogen $\left(\mathrm{H}_{2}\right)$ with molar mass 0.002 kilogram at a temperature of $70^{\circ} \mathrm{C}$ . Round to the nearest tenth of a meter per second.

A) $414.1 \mathrm{~m} / \mathrm{s}$
B) $2068.3 \mathrm{~m} / \mathrm{s}$
C) $1759.9 \mathrm{~m} / \mathrm{s}$
D) $934.4 \mathrm{~m} / \mathrm{s}$

Question 44

An oil platform is 7 miles off the coast, and a pipeline needs to run to a refinery that is 15 miles north of the platform along the coast. This requires that some of the pipeline will be offshore and some will run on land along the coast. If the pipeline comes on shore $x$ miles north of the platform, the underwater distance is given by $f(x)=\sqrt{49+x^{2}}$ , and the distance on land is given by $g(x)=15$ - $\mathrm{x}$ . The cost of the underwater pipeline is $\$ 500,000$ per mile, and the cost of the pipeline on land is $\$ 350,000$ per mile. If the pipeline comes on shore 6 miles north of the platform, what is the total cost of the pipeline? Do not round until the final step and then round to the nearest cent.

A) $\$ 7,759,772.23$
B) $\$ 6,709,772.23$
C) $\$ 7,287,817.78$
D) $\$ 7,726,840.56$

Question 45

The distance $d$ in miles that can be seen on the surface of the ocean is given by $d=1.2 \sqrt{h}$ , where $h$ is the height in feet above the surface. How high (to the nearest foot) would a platform have to be to see a distance of 16.5 miles?

A) $138 \mathrm{ft}$
B) $189 \mathrm{ft}$
C) $326 \mathrm{ft}$
D) $272 \mathrm{ft}$

Question 46

The radius of a sphere depends on its surface area, $S$ , and is given by the formula $r=\sqrt{\frac{\mathrm{S}}{4 \pi}}$
What is the surface area of a sphere with radius of 7.1 inches? Use 3.14 for $\pi$ . Round to the nearest tenth of a square inch.

A) 89.2 square inches
B) 22.3 square inches
C) 158.3 square inches
D) 633.1 square inches

Question 47

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 54.9, and the spring is to be stretched 4.3 feet from its natural length, how much work will be necessary? Round to the nearest tenth of a foot- pound.

A) 1015.1 foot- pounds
B) 118 foot- pounds
C) 507.6 foot-pounds
D) 56.9 foot- pounds

Question 48

Police use a formula $s=S \sqrt{\frac{l}{L}}$ , where Sis the test- car speed and $L$ is the test- skid length, to find the actual speed $\mathrm{s}$ in an accident which left a skid mark of 1 . Find the speed (nearest whole mph) when $\mathrm{S}=45 \mathrm{mph}, \mathrm{l}=150 \mathrm{ft}, \mathrm{L}=100 \mathrm{ft}$ .

A) $37 \mathrm{mph}$
B) $68 \mathrm{mph}$
C) $55 \mathrm{mph}$
D) $95 \mathrm{mph}$

Question 49

The length of the diagonal of a rectangle is given by $D=\sqrt{L^{2}+W^{2}}$ where $L$ and $W$ are the length and width of the rectangle. What is the length of the diagonal, D, of a rectangle that is 16 inches long and 2 inches wide? Round your answer to the nearest tenth of an inch, if necessary.

A) 15.9 inches
B) 5.7 inches
C) 4.2 inches
D) 16.1 inches

Question 50

The length of the diagonal of a box is given by $D=\sqrt{L^{2}+W^{2}+H^{2}}$ where $L, W$ , and $H$ are the length, width, and height of the box. Find the length of the diagonal, D, of a box that is $1 \mathrm{ft}$ long, $5 \mathrm{ft}$ high, and $4 \mathrm{ft}$ wide. Give the exact value.

A) $2 \sqrt{5} \mathrm{ft}$
B) $10 \mathrm{ft}$
C) $20 \mathrm{ft}$
D) $\sqrt{42} \mathrm{ft}$

Question 51

Find the domain of the function. Express the answer in interval notation.

- $f(x)=\sqrt{x+13}$

A) $(-\infty, 13)$
B) $[13, \infty)$
C) $(-\infty,-13)$
D) $[-13, \infty)$

Question 52

Find the domain of the function. Express the answer in interval notation.

- $g(x)=\sqrt{5 x^{2}+10}$

A) $(-2, \infty)$
B) $(-\infty, \infty)$
C) $[-2, \infty)$
D) $[0, \infty)$

Question 53

Find the domain of the function. Express the answer in interval notation.

- $f(x)=\sqrt{2 x+2}$

A) $[-1, \infty)$
B) $(-1, \infty)$
C) $(1, \infty)$
D) $(-\infty,-1)$

Question 54

Find the domain of the function. Express the answer in interval notation.

- $f(x)=\sqrt{9-2 x}$

A) $\left[\frac{9}{2}, \infty\right)$
B) $\left[-\infty, \frac{9}{2}\right]$
C) $\left[-\frac{9}{2}, \infty\right)$
D) $\left(-\infty, \frac{9}{2}\right)$

Question 55

Find the domain of the function. Express the answer in interval notation.

- $f(x)=\sqrt[6]{x+8}$

A) $(-\infty,-8)$
B) $(-\infty, 8)$
C) $[8, \infty)$
D) $[-8, \infty)$

Question 56

Find the domain of the function. Express the answer in interval notation.

- $f(x)=\sqrt[4]{x-19}$

A) $[19, \infty)$
B) $[0, \infty)$
C) $[-19, \infty)$
D) $(-\infty, \infty)$

Question 57

Find the domain of the function. Express the answer in interval notation.

- $f(x)=\sqrt[3]{x-18}$

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

Question 58

Find the domain of the function. Express the answer in interval notation.

- $f(x)=3-5 \sqrt[4]{3 t+30}$

A) $[-10, \infty)$
B) $(-\infty, \infty)$
C) $(-\infty,-10)$
D) $[10, \infty)$

Question 59

Find the domain of the function. Express the answer in interval notation.

- $f(x)=5+2 \sqrt[6]{3 t-42}$

A) $[14, \infty)$
B) $(-\infty, 14)$
C) $(-\infty, \infty)$
D) $[-14, \infty)$

Question 60

Find the domain of the function. Express the answer in interval notation.

- $f(x)=\sqrt[5]{2 x-4}$

A) $[2, \infty)$
B) $(-\infty, 2)$
C) $(-\infty, \infty)$
D) $[-2, \infty)$

Question 61

How many real square roots does any negative number have?

A) It depends on the number
B) None
C) One
D) Two

Question 62

How many real cube roots does any negative number have?

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

Question 63

If $n$ is odd, under what condition is $\sqrt[n]{a}$ negative?

A) When a is negative
B) When a is zero
C) When a is positive
D) Never

Question 64

Explain in your own words why $\sqrt{\mathrm{a}^{2}} \neq \mathrm{a}$ when a is negative.

Question 65

Explain why $\sqrt[k]{\mathrm{x}^{\mathrm{k}}}=|\mathrm{x}|$ when $\mathrm{k}$ is even but $\sqrt[k]{\mathrm{x}^{\mathrm{k}}}=\mathrm{x}$ when $\mathrm{k}$ is odd.

Question 66

Rewrite the radical expression using rational exponents. A ssume all variables represent nonnegative real numbers.

- $\sqrt[4]{17}$

A) $-171 / 4$
B) $171 / 4$
C) $\frac{1}{17^{4}}$
D) $4^{1 / 17}$

Question 67

Rewrite the radical expression using rational exponents. A ssume all variables represent nonnegative real numbers.

- $\sqrt[6]{\mathrm{mn}}$

A) $\frac{1}{(\mathrm{mn})^{6}}$
B) $(\mathrm{mn})^{1 / 6}$
C) $m^{1 / 6} n$
D) $\mathrm{mn}^{1 / 6}$

Question 68

Rewrite the radical expression using rational exponents. A ssume all variables represent nonnegative real numbers.

- $\sqrt{\mathrm{x}^{9}}$

A) $x 9 / 2$
B) $-x^{2 / 9}$
C) $x^{-2 / 9}$
D) $x^{-9 / 2}$

Question 69

Rewrite the radical expression using rational exponents. A ssume all variables represent nonnegative real numbers.

- $(\sqrt[9]{b})^{7}$

A) $b^{7 / 9}$
B) $b^{-9 / 7}$
C) $b^{-7 / 9}$
D) $b^{9 / 7}$

Question 70

Rewrite the radical expression using rational exponents. A ssume all variables represent nonnegative real numbers.

- $\sqrt[8]{m^{7}}$

A) $m^{7 / 8}$
B) $\frac{1}{(m n)^{8}}$
C) $-m^{7 / 8}$
D) $m^{8 / 7}$

Question 71

Rewrite the radical expression using rational exponents. A ssume all variables represent nonnegative real numbers.

- $\left(\sqrt[5]{5 x^{3} y}\right)^{2}$

A) $(5 x y)^{6 / 5}$
B) $\left(5 x^{3} y\right)^{5 / 2}$
C) $\left(5 x^{3} y\right)^{2 / 5}$
D) $\left(5 x^{3} y\right)^{10}$

Question 72

Rewrite the radical expression using rational exponents. A ssume all variables represent nonnegative real numbers.

- $\sqrt[2]{x^{5} y^{7}}$

A) $\left(x^{5} y z^{7}\right)^{1 / 2}$
B) $x^{3} y^{-1} z^{5}$
C) $\left(x^{5} y z^{7}\right)^{2}$
D) $\left(x^{5} y z^{7}\right)^{-2}$

Question 73

Rewrite the radical expression using rational exponents. A ssume all variables represent nonnegative real numbers.

- $\left(\sqrt[9]{5 x^{4} y}\right)^{7}$

A) $\left(5 x^{4} y\right)^{7 / 9}$
B) $(5 x y)^{19}$
C) $\left(5 x^{4} y\right)^{9 / 7}$
D) $\left(5 x^{4} y\right)^{2}$

Question 74

Rewrite as a radical expression and simplify if possible. Assume all variables represent nonnegative real numbers.

- $400^{1 / 2}$

A) 80
B) 40
C) 10
D) 20

Question 75

Rewrite as a radical expression and simplify if possible. Assume all variables represent nonnegative real numbers.

- $343^{1 / 3}$

A) 7
B) 2401
C) 21
D) 7203

Question 76

Rewrite as a radical expression and simplify if possible. Assume all variables represent nonnegative real numbers.

- $(-36)^{1 / 2}$

A) 6
B) -3
C) -6
D) Not a real number

Question 77

Rewrite as a radical expression and simplify if possible. Assume all variables represent nonnegative real numbers.

- $\left(y^{20}\right)^{1 / 4}$

A) $y^{20}$
B) $7 \mathrm{y}$
C) $y^{5}$
D) $y^{7}$

Question 78

Rewrite as a radical expression and simplify if possible. Assume all variables represent nonnegative real numbers.

- $\left(x^{4} y^{6}\right)^{1 / 2}$

A) $x^{2} y^{3}$
B) $2 x^{4} y^{6}$
C) $x^{8} y^{12}$
D) $x^{3} y^{2}$

Question 79

Rewrite as a radical expression and simplify if possible. Assume all variables represent nonnegative real numbers.

- $\left(169 r^{14}\right)^{1 / 2}$

A) $\sqrt{169 r^{14}}$
B) $13 r^{7}$
C) $13 \sqrt{r^{14}}$
D) $13 r^{14}$

Question 80

Rewrite as a radical expression and simplify if possible. Assume all variables represent nonnegative real numbers.

- $\left(4 x^{4} y^{4}\right)^{1 / 2}$

A) $2 x^{4} y^{2}$
B) $2 x^{2} y$
C) $2 x^{2} y^{2}$
D) $x^{2} y^{2}$