Intermediate Algebra : Concepts and Applications

Marvin L. Bittinger, David J. Ellenbogen, Barbara L. Johnson

Chapter 7

Exponents and Radicals - all with Video Answers

Educators

AG

Section 1

Radical Expressions and Functions

Problem 1

Two or three words appear right-side of the blank. Choose the correct word to complete the statement.
Every positive number has _____(one/two) $\operatorname{root}(\mathrm{s})$.

AG

Ankit G.

Numerade Educator

Problem 2

Two or three words appear right-side of the blank. Choose the correct word to complete the statement.
The principal square root is never _____(negative/positive).

AG

Ankit G.

Numerade Educator

Problem 3

Two or three words appear right-side of the blank. Choose the correct word to complete the statement.
Even if $a$ represents a negative number, $\sqrt{a^{2}}$ is _____(negative/positive).

AG

Ankit G.

Numerade Educator

Problem 4

Two or three words appear right-side of the blank. Choose the correct word to complete the statement.
If $a$ represents a _____(negative/positive) number, then $\sqrt{a^{2}}=-a$.

AG

Ankit G.

Numerade Educator

Problem 5

Two or three words appear right-side of the blank. Choose the correct word to complete the statement.
If $a$ is a whole number that is not a perfect square, then $\sqrt{a}$ is a(n) _____(irrational/rational) number.

AG

Ankit G.

Numerade Educator

Problem 6

Two or three words appear right-side of the blank. Choose the correct word to complete the statement.
The domain of the function $f$ given by $f(x)=\sqrt[3]{x}$ is the set of all _____(whole/real/positive) numbers.

AG

Ankit G.

Numerade Educator

Problem 7

Two or three words appear right-side of the blank. Choose the correct word to complete the statement.
If $\sqrt[4]{x}$ is a real number, then $x$ must be _____(negative/positive/nonnegative).

AG

Ankit G.

Numerade Educator

Problem 8

Two or three words appear right-side of the blank. Choose the correct word to complete the statement.
If $\sqrt[3]{x}$ is negative, then $x$ must be _____(negative/positive).

AG

Ankit G.

Numerade Educator

Problem 9

Find all of its square roots.
$$
64
$$

AG

Ankit G.

Numerade Educator

Problem 10

Find all of its square roots.
$$
81
$$

AG

Ankit G.

Numerade Educator

Problem 11

Find all of its square roots.
$$
100
$$

AG

Ankit G.

Numerade Educator

Problem 12

Find all of its square roots.
$$
121
$$

AG

Ankit G.

Numerade Educator

Problem 13

Find all of its square roots.
$$
400
$$

AG

Ankit G.

Numerade Educator

Problem 14

Find all of its square roots.
$$
2500
$$

AG

Ankit G.

Numerade Educator

Problem 15

Find all of its square roots.
$$
625
$$

AG

Ankit G.

Numerade Educator

Problem 16

Find all of its square roots.
$$
225
$$

AG

Ankit G.

Numerade Educator

Problem 17

Simplify.
$$
\sqrt{49}
$$

Julie S.

Numerade Educator

Problem 18

Simplify.
$$
\sqrt{144}
$$

Julie S.

Numerade Educator

Problem 19

Simplify.
$$
-\sqrt{16}
$$

Cory K.

Numerade Educator

Problem 20

Simplify.
$$
-\sqrt{100}
$$

Cory K.

Numerade Educator

Problem 21

Simplify.
$$
\sqrt{\frac{36}{49}}
$$

Cory K.

Numerade Educator

Problem 22

Simplify.
$$
\sqrt{\frac{4}{9}}
$$

Cory K.

Numerade Educator

Problem 23

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

Cory K.

Numerade Educator

Problem 24

Simplify.
$$
-\sqrt{\frac{81}{144}}
$$

Cory K.

Numerade Educator

Problem 25

Simplify.
$$
\sqrt{0.04}
$$

Cory K.

Numerade Educator

Problem 26

Simplify.
$$
\sqrt{0.36}
$$

Cory K.

Numerade Educator

Problem 27

Simplify.
$$
\sqrt{0.0081}
$$

Cory K.

Numerade Educator

Problem 28

Simplify.
$$
\sqrt{0.0016}
$$

AB

Andy B.

Numerade Educator

Problem 29

Find the specified function value, if it exists.
$$
f(t)=\sqrt{5 t-10} ; f(3), f(2), f(1), f(-1)
$$

AG

Ankit G.

Numerade Educator

Problem 30

Find the specified function value, if it exists.
$$
g(x)=\sqrt{x^{2}-25} ; g(-6), g(3), g(6), g(13)
$$

AG

Ankit G.

Numerade Educator

Problem 31

Find the specified function value, if it exists.
$$
t(x)=-\sqrt{2 x^{2}-1} ; t(5), t(0), t(-1), t\left(-\frac{1}{2}\right)
$$

AG

Ankit G.

Numerade Educator

Problem 32

Find the specified function value, if it exists.
$$
p(z)=\sqrt{2 z-20} ; p(4), p(10), p(12), p(0)
$$

AG

Ankit G.

Numerade Educator

Problem 33

Find the specified function value, if it exists.
$$
f(t)=\sqrt{t^{2}+1} ; f(0), f(-1), f(-10)
$$

AG

Ankit G.

Numerade Educator

Problem 34

Find the specified function value, if it exists.
$$
g(x)=-\sqrt{(x+1)^{2}} ; g(-3), g(4), g(-5)
$$

AG

Ankit G.

Numerade Educator

Problem 35

Simplify. Variables may represent any real number, so remember to use absolute-value notation when necessary. If a root cannot be simplified, state this.
$$
\sqrt{100 x^{2}}
$$

AG

Ankit G.

Numerade Educator

Problem 36

Simplify. Variables may represent any real number, so remember to use absolute-value notation when necessary. If a root cannot be simplified, state this.
$$
\sqrt{16 t^{2}}
$$

AG

Ankit G.

Numerade Educator

Problem 37

Simplify. Variables may represent any real number, so remember to use absolute-value notation when necessary. If a root cannot be simplified, state this.
$$
\sqrt{(-4 b)^{2}}
$$

AG

Ankit G.

Numerade Educator

Problem 38

Simplify. Variables may represent any real number, so remember to use absolute-value notation when necessary. If a root cannot be simplified, state this.
$$
\sqrt{(-7 c)^{2}}
$$

AG

Ankit G.

Numerade Educator

Problem 39

Simplify. Variables may represent any real number, so remember to use absolute-value notation when necessary. If a root cannot be simplified, state this.
$$
\sqrt{(8-t)^{2}}
$$

AG

Ankit G.

Numerade Educator

Problem 40

Simplify. Variables may represent any real number, so remember to use absolute-value notation when necessary. If a root cannot be simplified, state this.
$$
\sqrt{(a+3)^{2}}
$$

AG

Ankit G.

Numerade Educator

Problem 41

Simplify. Variables may represent any real number, so remember to use absolute-value notation when necessary. If a root cannot be simplified, state this.
$$
\sqrt{y^{2}+16 y+64}
$$

AG

Ankit G.

Numerade Educator

Problem 42

Simplify. Variables may represent any real number, so remember to use absolute-value notation when necessary. If a root cannot be simplified, state this.
$$
\sqrt{x^{2}-4 x+4}
$$

AG

Ankit G.

Numerade Educator

Problem 43

Simplify. Variables may represent any real number, so remember to use absolute-value notation when necessary. If a root cannot be simplified, state this.
$$
\sqrt{4 x^{2}+28 x+49}
$$

AG

Ankit G.

Numerade Educator

Problem 44

Simplify. Variables may represent any real number, so remember to use absolute-value notation when necessary. If a root cannot be simplified, state this.
$$
\sqrt{9 x^{2}-30 x+25}
$$

AG

Ankit G.

Numerade Educator

Problem 45

Simplify. Variables may represent any real number, so remember to use absolute-value notation when necessary. If a root cannot be simplified, state this.
$$
\sqrt{a^{22}}
$$

AG

Ankit G.

Numerade Educator

Problem 46

Simplify. Variables may represent any real number, so remember to use absolute-value notation when necessary. If a root cannot be simplified, state this.
$$
\sqrt{x^{10}}
$$

AG

Ankit G.

Numerade Educator

Problem 47

Simplify. Variables may represent any real number, so remember to use absolute-value notation when necessary. If a root cannot be simplified, state this.
$$
\sqrt{-25}
$$

AG

Ankit G.

Numerade Educator

Problem 48

Simplify. Variables may represent any real number, so remember to use absolute-value notation when necessary. If a root cannot be simplified, state this.
$$
\sqrt{-16}
$$

AG

Ankit G.

Numerade Educator

Problem 49

Simplify.
$$
\sqrt[3]{-1}
$$

AG

Ankit G.

Numerade Educator

Problem 50

Simplify.
$$
-\sqrt[3]{-1000}
$$

AG

Ankit G.

Numerade Educator

Problem 51

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

Amy J.

Numerade Educator

Problem 52

Simplify.
$$
\sqrt[3]{27}
$$

Chandler A.

Numerade Educator

Problem 53

Simplify.
$$
-\sqrt[3]{-125 y^{3}}
$$

AG

Ankit G.

Numerade Educator

Problem 54

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

AG

Ankit G.

Numerade Educator

Problem 55

Identify the radicand and the index for each expression.
$$
5 \sqrt{p^{2}+4}
$$

Cory K.

Numerade Educator

Problem 56

Identify the radicand and the index for each expression.
$$
-7 \sqrt{y^{2}-8}
$$

Cory K.

Numerade Educator

Problem 57

Identify the radicand and the index for each expression.
$$
x^{2} y^{3} \sqrt[5]{\frac{x}{y+4}}
$$

AG

Ankit G.

Numerade Educator

Problem 58

Identify the radicand and the index for each expression.
$$
\frac{a^{2}}{b} \sqrt[6]{a(a+b)}
$$

AG

Ankit G.

Numerade Educator

Problem 59

Simplify. Use absolute-value notation when necessary.
$$
-\sqrt[4]{256}
$$

AG

Ankit G.

Numerade Educator

Problem 60

Simplify. Use absolute-value notation when necessary.
$$
-\sqrt[4]{625}
$$

AG

Ankit G.

Numerade Educator

Problem 61

Simplify. Use absolute-value notation when necessary.
$$
-\sqrt[5]{-\frac{32}{243}}
$$

AG

Ankit G.

Numerade Educator

Problem 62

Simplify. Use absolute-value notation when necessary.
$$
\sqrt[5]{-\frac{1}{32}}
$$

AG

Ankit G.

Numerade Educator

Problem 63

Simplify. Use absolute-value notation when necessary.
$$
\sqrt[6]{x^{6}}
$$

AG

Ankit G.

Numerade Educator

Problem 64

Simplify. Use absolute-value notation when necessary.
$$
\sqrt[8]{y^{8}}
$$

AG

Ankit G.

Numerade Educator

Problem 65

Simplify. Use absolute-value notation when necessary.
$$
\sqrt[9]{t^{9}}
$$

AG

Ankit G.

Numerade Educator

Problem 66

Simplify. Use absolute-value notation when necessary.
$$
\sqrt[5]{a^{5}}
$$

AG

Ankit G.

Numerade Educator

Problem 67

Simplify. Use absolute-value notation when necessary.
$$
\sqrt[4]{(6 a)^{4}}
$$

AG

Ankit G.

Numerade Educator

Problem 68

Simplify. Use absolute-value notation when necessary.
$$
\sqrt[4]{(7 b)^{4}}
$$

AG

Ankit G.

Numerade Educator

Problem 69

Simplify. Use absolute-value notation when necessary.
$$
\sqrt[10]{(-6)^{10}}
$$

AG

Ankit G.

Numerade Educator

Problem 70

Simplify. Use absolute-value notation when necessary.
$$
\sqrt[12]{(-10)^{12}}
$$

AG

Ankit G.

Numerade Educator

Problem 71

Simplify. Use absolute-value notation when necessary.
$$
\sqrt[414]{(a+b)^{414}}
$$

AG

Ankit G.

Numerade Educator

Problem 72

Simplify. Use absolute-value notation when necessary.
$$
\sqrt[1976]{(2 a+b)^{1976}}
$$

AG

Ankit G.

Numerade Educator

Problem 73

Simplify. Assume that no radicands were formed by raising negative quantities to even powers. Thus absolute-value notation is not necessary.
$$
\sqrt{16 x^{2}}
$$

AG

Ankit G.

Numerade Educator

Problem 74

Simplify. Assume that no radicands were formed by raising negative quantities to even powers. Thus absolute-value notation is not necessary.
$$
\sqrt{25 t^{2}}
$$

AG

Ankit G.

Numerade Educator

Problem 75

Simplify. Assume that no radicands were formed by raising negative quantities to even powers. Thus absolute-value notation is not necessary.
$$
-\sqrt{(3 t)^{2}}
$$

AG

Ankit G.

Numerade Educator

Problem 76

Simplify. Assume that no radicands were formed by raising negative quantities to even powers. Thus absolute-value notation is not necessary.
$$
-\sqrt{(7 c)^{2}}
$$

AG

Ankit G.

Numerade Educator

Problem 77

Simplify. Assume that no radicands were formed by raising negative quantities to even powers. Thus absolute-value notation is not necessary.
$$
\sqrt{(-5 b)^{2}}
$$

AG

Ankit G.

Numerade Educator

Problem 78

Simplify. Assume that no radicands were formed by raising negative quantities to even powers. Thus absolute-value notation is not necessary.
$$
\sqrt{(-10 a)^{2}}
$$

AG

Ankit G.

Numerade Educator

Problem 79

Simplify. Assume that no radicands were formed by raising negative quantities to even powers. Thus absolute-value notation is not necessary.
$$
\sqrt{a^{2}+2 a+1}
$$

AG

Ankit G.

Numerade Educator

Problem 80

Simplify. Assume that no radicands were formed by raising negative quantities to even powers. Thus absolute-value notation is not necessary.
$$
\sqrt{9-6 y+y^{2}}
$$

AG

Ankit G.

Numerade Educator

Problem 81

Simplify. Assume that no radicands were formed by raising negative quantities to even powers. Thus absolute-value notation is not necessary.
$$
\sqrt[4]{16 x^{4}}
$$

AG

Ankit G.

Numerade Educator

Problem 82

Simplify. Assume that no radicands were formed by raising negative quantities to even powers. Thus absolute-value notation is not necessary.
$$
\sqrt[4]{81 x^{4}}
$$

AG

Ankit G.

Numerade Educator

Problem 83

Simplify. Assume that no radicands were formed by raising negative quantities to even powers. Thus absolute-value notation is not necessary.
$$
\sqrt[3]{(x-1)^{3}}
$$

AG

Ankit G.

Numerade Educator

Problem 84

Simplify. Assume that no radicands were formed by raising negative quantities to even powers. Thus absolute-value notation is not necessary.
$$
-\sqrt[3]{(7 y)^{3}}
$$

AG

Ankit G.

Numerade Educator

Problem 85

Simplify. Assume that no radicands were formed by raising negative quantities to even powers. Thus absolute-value notation is not necessary.
$$
\sqrt{t^{18}}
$$

AG

Ankit G.

Numerade Educator

Problem 86

Simplify. Assume that no radicands were formed by raising negative quantities to even powers. Thus absolute-value notation is not necessary.
$$
\sqrt{a^{14}}
$$

AG

Ankit G.

Numerade Educator

Problem 87

Simplify. Assume that no radicands were formed by raising negative quantities to even powers. Thus absolute-value notation is not necessary.
$$
\sqrt{(x-2)^{8}}
$$

AG

Ankit G.

Numerade Educator

Problem 88

Simplify. Assume that no radicands were formed by raising negative quantities to even powers. Thus absolute-value notation is not necessary.
$$
\sqrt{(x+3)^{10}}
$$

AG

Ankit G.

Numerade Educator

Problem 89

Find the specified function value, if it exists.
$$
f(x)=\sqrt[3]{x+1} ; f(7), f(26), f(-9), f(-65)
$$

AG

Ankit G.

Numerade Educator

Problem 90

Find the specified function value, if it exists.
$$
g(x)=-\sqrt[3]{2 x-1} ; g(0), g(-62), g(-13), g(63)
$$

AG

Ankit G.

Numerade Educator

Problem 91

Find the specified function value, if it exists.
$$
g(t)=\sqrt[4]{t-3} ; g(19), g(-13), g(1), g(84)
$$

AG

Ankit G.

Numerade Educator

Problem 92

Find the specified function value, if it exists.
$$
f(t)=\sqrt[4]{t+1} ; f(0), f(15), f(-82), f(80)
$$

AG

Ankit G.

Numerade Educator

Problem 93

Determine the domain of each function described.
$$
f(x)=\sqrt{x-6}
$$

Cory K.

Numerade Educator

Problem 94

Determine the domain of each function described.
$$
g(x)=\sqrt{x+8}
$$

Cory K.

Numerade Educator

Problem 95

Determine the domain of each function described.
$$
g(t)=\sqrt[4]{t+8}
$$

Cory K.

Numerade Educator

Problem 96

Determine the domain of each function described.
$$
f(x)=\sqrt[4]{x-9}
$$

Cory K.

Numerade Educator

Problem 97

Determine the domain of each function described.
$$
g(x)=\sqrt[4]{10-2 x}
$$

AG

Ankit G.

Numerade Educator

Problem 98

Determine the domain of each function described.
$$
g(t)=\sqrt[3]{2 t-6}
$$

Cory K.

Numerade Educator

Problem 99

Determine the domain of each function described.
$$
f(t)=\sqrt[5]{2 t+7}
$$

AG

Ankit G.

Numerade Educator

Problem 100

Determine the domain of each function described.
$$
f(t)=\sqrt[6]{4+3 t}
$$

AG

Ankit G.

Numerade Educator

Problem 101

Determine the domain of each function described.
$$
h(z)=-\sqrt[6]{5 z+2}
$$

Cory K.

Numerade Educator

Problem 102

Determine the domain of each function described.
$$
d(x)=-\sqrt[4]{5-7 x}
$$

AG

Ankit G.

Numerade Educator

Problem 103

Determine the domain of each function described.
$$
f(t)=7+\sqrt[8]{t^{8}}
$$

Cory K.

Numerade Educator

Problem 104

Determine the domain of each function described.
$$
g(t)=9+\sqrt[6]{t^{6}}
$$

Cory K.

Numerade Educator

Problem 105

Explain how to write the negative square root of a number using radical notation.

Cory K.

Numerade Educator

Problem 106

Does the square root of a number's absolute value always exist? Why or why not?

Cory K.

Numerade Educator

Problem 107

Let $f(x)=3 x-1$ and $g(x)=\frac{1}{x}$.
$$
\text { Find } f\left(\frac{1}{3}\right)
$$

AG

Ankit G.

Numerade Educator

Problem 108

Let $f(x)=3 x-1$ and $g(x)=\frac{1}{x}$.
Find the domain of $f$.

AG

Ankit G.

Numerade Educator

Problem 109

Let $f(x)=3 x-1$ and $g(x)=\frac{1}{x}$.
Find the domain of $g$.

AG

Ankit G.

Numerade Educator

Problem 110

Let $f(x)=3 x-1$ and $g(x)=\frac{1}{x}$.
Find $(f+g)(x)$.

AG

Ankit G.

Numerade Educator

Problem 111

Let $f(x)=3 x-1$ and $g(x)=\frac{1}{x}$.
Find $(f g)(x)$.

AG

Ankit G.

Numerade Educator

Problem 112

Let $f(x)=3 x-1$ and $g(x)=\frac{1}{x}$.
Which function, $f$ or $g,$ is a linear function?

AG

Ankit G.

Numerade Educator

Problem 113

Under what conditions does the $n$ th root of $x^{3}$ exist as a real number? Explain your reasoning.

AG

Ankit G.

Numerade Educator

Problem 114

Describe the conditions for which absolutevalue notation is used when simplifying a radical expression.

AG

Ankit G.

Numerade Educator

Problem 115

Firefighting. The number of gallons per minute discharged from a fire hose depends on the diameter of the nozzle and the nozzle pressure. The following graph illustrates the amount of water flow for a 2 -in. diameter solid bore nozzle at various nozzle pressures. (GRAPH CANNOT COPY)
The water flow in the graph can be modeled by
$$
f(p)=118.8 \sqrt{p}
$$
where $p$ is the nozzle pressure, in pounds per square inch (psi), and $f(p)$ is the water flow, in gallons per minute $(G P M) .$ Use this function. Estimate the water flow when the nozzle pressure is 50 psi.

AG

Ankit G.

Numerade Educator

Problem 116

Firefighting. The number of gallons per minute discharged from a fire hose depends on the diameter of the nozzle and the nozzle pressure. The following graph illustrates the amount of water flow for a 2 -in. diameter solid bore nozzle at various nozzle pressures. (GRAPH CANNOT COPY)
The water flow in the graph can be modeled by
$$
f(p)=118.8 \sqrt{p}
$$
where $p$ is the nozzle pressure, in pounds per square inch (psi), and $f(p)$ is the water flow, in gallons per minute $(G P M) .$ Use this function. Estimate the water flow when the nozzle pressure is 175 psi.

AG

Ankit G.

Numerade Educator

Problem 117

The number of species $S$ of plants in Guyana in an area of $A$ hectares can be estimated using the formula
$$
S=88.63 \sqrt[4]{A}
$$
The Kaieteur National Park in Guyana has an area of $63,000$ hectares. How many species of plants are in the park?
Data: Hans ter Steege, "A Perspective on Guyana and its Plant Richness," as found on www.bio.uu.nl

AG

Ankit G.

Numerade Educator

Problem 118

A parking lot has attendants to park the cars. The number of spaces $N$ needed for waiting cars before attendants can get to them is given by the formula $N=2.5 \sqrt{A},$ where $A$ is the number of arrivals in peak hours. Find the number of spaces needed for the given number of arrivals in peak hours: (a) $25 ;$ (b) 36 (c) $49 ;$ (d) 64.

AG

Ankit G.

Numerade Educator

Problem 119

Determine the domain of each function described. Then draw the graph of each function.
$$
f(x)=\sqrt{x+5}
$$

Cory K.

Numerade Educator

Problem 120

Determine the domain of each function described. Then draw the graph of each function.
$$
g(x)=\sqrt{x}+5
$$

Cory K.

Numerade Educator

Problem 121

Determine the domain of each function described. Then draw the graph of each function.
$$
g(x)=\sqrt{x}-2
$$

Cory K.

Numerade Educator

Problem 122

Determine the domain of each function described. Then draw the graph of each function.
$$
f(x)=\sqrt{x-2}
$$

Cory K.

Numerade Educator

Problem 123

Find the domain of $f$ if
$$
f(x)=\frac{\sqrt{x+3}}{\sqrt[4]{2-x}}
$$

AG

Ankit G.

Numerade Educator

Problem 124

Find the domain of $g$ if
$$
g(x)=\frac{\sqrt[4]{5-x}}{\sqrt[6]{x+4}}
$$

AG

Ankit G.

Numerade Educator

Problem 125

Find the domain of $F$ if
$$
F(x)=\frac{x}{\sqrt{x^{2}-5 x-6}}
$$

AG

Ankit G.

Numerade Educator

Problem 126

Use a graphing calculator to check your answers to Exercises $39,45,$ and $67 .$ On some graphing calculators, a MATH key is needed to enter higher roots.

AG

Ankit G.

Numerade Educator

Problem 127

John Tierney and Garth Sundem have developed an equation that can be used to estimate the probability that a celebrity marriage will last. The probability $P$ that a couple will still be married in $T$ years is given by
$$
P=50 \sqrt[15]{\frac{\mathrm{NYT}(\mathrm{Ah}+\mathrm{Aw})}{\mathrm{ENQ}(\mathrm{Sc}+5)} \cdot \mathrm{Md} \cdot\left[\frac{\mathrm{Md}}{(\mathrm{Md}+2)}\right]^{T^{2}}}
$$
where $\mathrm{NYT}=$ the number of times since 1990 that the wife's name appeared in the New York Times, ENQ $=$ the number of times since 1990 that the wife's name appeared in the National Enquirer, $\mathrm{Ah}=$ the husband's age, in years, $\mathrm{Aw}=$ the wife's age, in years, $\mathrm{Md}=$ the number of months that the couple dated before marriage, and $\mathrm{Sc}=$ the number of the top five photos returned by a Google images search for the wife's name in which she was "scantily clad." Kate Middleton and Prince William were both 29 when they were married and had dated for 120 months. Kate appeared 258 times in the $N e w$ York Times and 44 times in the National Enquirer, and her Sc value was $0 .$ What is the probability that the marriage will last 5 years? Data: "Refining the Formula That Predicts Celebrity Marriages' Doom," www.nytimes.com, March $12,2012$

AG

Ankit G.

Numerade Educator