Algebra 2 and Trigonometry

Ann Xavier Gantert

Chapter 3

REAL NUMBERS AND RADICALS - all with Video Answers

Educators

Section 2

Roots and Radicals

Problem 1

a. Kevin said that if the index of a radical is even and the radicand is positive, then the radical has two real roots. Do you agree with Kevin? Explain why or why not.
b. Kevin said that if the index of a radical is odd, then the radical has one real root and that the root is positive if the radicand is positive and negative if the radicand is negative. Do you agree with Kevin? Explain why or why not.

James K.

James K.

Numerade Educator

Problem 2

a. Sarah said that in the set of real numbers, $\sqrt{a}$ is one of the two equal factors whose product is $a .$ Therefore, $\sqrt{a} \cdot \sqrt{a}=a$ for some values of $a$ . Do you agree with Sarah? Explain
why or why not.
b. If you agree with Sarah, for which values of $a$ is the statement true? Explain.

Dilip P.

Numerade Educator

Problem 3

In $3-10,$ tell whether each represents a number that is rational, irrational, or neither.
$$
\sqrt{25}
$$

James K.

Numerade Educator

Problem 4

In $3-10,$ tell whether each represents a number that is rational, irrational, or neither.
$$
\sqrt{8}
$$

Dilip P.

Numerade Educator

Problem 5

In $3-10,$ tell whether each represents a number that is rational, irrational, or neither.
$$
\sqrt{-8}
$$

James K.

Numerade Educator

Problem 6

In $3-10,$ tell whether each represents a number that is rational, irrational, or neither.
$$
\sqrt[3]{-8}
$$

Dilip P.

Numerade Educator

Problem 7

In $3-10,$ tell whether each represents a number that is rational, irrational, or neither.
$$
\sqrt{0}
$$

James K.

Numerade Educator

Problem 8

In $3-10,$ tell whether each represents a number that is rational, irrational, or neither.
$$
\sqrt[4]{16}
$$

Dilip P.

Numerade Educator

Problem 9

In $3-10,$ tell whether each represents a number that is rational, irrational, or neither.
$$
\sqrt[5]{-243}
$$

James K.

Numerade Educator

Problem 10

In $3-10,$ tell whether each represents a number that is rational, irrational, or neither.
$$
\sqrt{0.25}
$$

Dilip P.

Numerade Educator

Problem 11

In $11-38,$ evaluate each expression in the set of real numbers.
$$
\sqrt{16}
$$

James K.

Numerade Educator

Problem 12

In $11-38,$ evaluate each expression in the set of real numbers.
$$
\pm \sqrt{16}
$$

Dilip P.

Numerade Educator

Problem 13

In $11-38,$ evaluate each expression in the set of real numbers.
$$
-\sqrt{16}
$$

James K.

Numerade Educator

Problem 14

In $11-38,$ evaluate each expression in the set of real numbers.
$$
\sqrt{625}
$$

Dilip P.

Numerade Educator

Problem 15

In $11-38,$ evaluate each expression in the set of real numbers.
$$
\sqrt{169}
$$

James K.

Numerade Educator

Problem 16

In $11-38,$ evaluate each expression in the set of real numbers.
$$
-\sqrt{0.04}
$$

Dilip P.

Numerade Educator

Problem 17

In $11-38,$ evaluate each expression in the set of real numbers.
$$
\pm \sqrt{0.64}
$$

James K.

Numerade Educator

Problem 18

In $11-38,$ evaluate each expression in the set of real numbers.
$$
\sqrt{1.44}
$$

Dilip P.

Numerade Educator

Problem 19

In $11-38,$ evaluate each expression in the set of real numbers.
$$
\sqrt[3]{27}
$$

James K.

Numerade Educator

Problem 20

In $11-38,$ evaluate each expression in the set of real numbers.
$$
\sqrt[4]{16}
$$

Dilip P.

Numerade Educator

Problem 21

In $11-38,$ evaluate each expression in the set of real numbers.
$$
\sqrt[3]{-125}
$$

James K.

Numerade Educator

Problem 22

In $11-38,$ evaluate each expression in the set of real numbers.
$$
-\sqrt[3]{125}
$$

Dilip P.

Numerade Educator

Problem 23

In $11-38,$ evaluate each expression in the set of real numbers.
$$
-\sqrt[3]{-125}
$$

James K.

Numerade Educator

Problem 24

In $11-38,$ evaluate each expression in the set of real numbers.
$$
\sqrt[5]{-1}
$$

Dilip P.

Numerade Educator

Problem 25

In $11-38,$ evaluate each expression in the set of real numbers.
$$
\sqrt{\frac{4}{25}}
$$

James K.

Numerade Educator

Problem 26

In $11-38,$ evaluate each expression in the set of real numbers.
$$
-\sqrt{\frac{49}{36}}
$$

Dilip P.

Numerade Educator

Problem 27

In $11-38,$ evaluate each expression in the set of real numbers.
$$
\sqrt[3]{\frac{8}{27}}
$$

James K.

Numerade Educator

Problem 28

In $11-38,$ evaluate each expression in the set of real numbers.
$$
\sqrt[5]{-\frac{1}{32}}
$$

Dilip P.

Numerade Educator

Problem 29

In $11-38,$ evaluate each expression in the set of real numbers.
$$
\sqrt[3]{0.001}
$$

James K.

Numerade Educator

Problem 30

In $11-38,$ evaluate each expression in the set of real numbers.
$$
\sqrt[4]{0.0256}
$$

Dilip P.

Numerade Educator

Problem 31

In $11-38,$ evaluate each expression in the set of real numbers.
$$
\sqrt{x^{6}}, x \geq 0
$$

James K.

Numerade Educator

Problem 32

In $11-38,$ evaluate each expression in the set of real numbers.
$$
\sqrt{100 c^{4}}, c \geq 0
$$

Dilip P.

Numerade Educator

Problem 33

In $11-38,$ evaluate each expression in the set of real numbers.
$$
\sqrt{0.25 x^{2}}, x \geq 0
$$

James K.

Numerade Educator

Problem 34

In $11-38,$ evaluate each expression in the set of real numbers.
$$
\sqrt[3]{\frac{1,000}{a^{3}},} a \neq 0
$$

Dilip P.

Numerade Educator

Problem 35

In $11-38,$ evaluate each expression in the set of real numbers.
$$
-\sqrt[4]{\frac{b^{8}}{1,296}}, b \geq 0
$$

James K.

Numerade Educator

Problem 36

In $11-38,$ evaluate each expression in the set of real numbers.
$$
\sqrt[5]{-0.00001 y^{5}}, y \geq 0
$$

Dilip P.

Numerade Educator

Problem 37

In $11-38,$ evaluate each expression in the set of real numbers.
$$
\sqrt[k]{1}, k \text { an integer greater than } 2
$$

James K.

Numerade Educator

Problem 38

In $11-38,$ evaluate each expression in the set of real numbers.
$$
\sqrt[k]{x^{2 k}}, k \text { an integer greater than } 2
$$

Dilip P.

Numerade Educator

Problem 39

In $39-42,$ find the set of real numbers for which the given radical is a real number.
$$
\sqrt{x-2}
$$

James K.

Numerade Educator

Problem 40

In $39-42,$ find the set of real numbers for which the given radical is a real number.
$$
\sqrt{9-3 x}
$$

Dilip P.

Numerade Educator

Problem 41

In $39-42,$ find the set of real numbers for which the given radical is a real number.
$$
\sqrt{4 x+12}
$$

James K.

Numerade Educator

Problem 42

In $39-42,$ find the set of real numbers for which the given radical is a real number.
$$
\sqrt[4]{x+5}
$$

Dilip P.

Numerade Educator

Problem 43

In $43-46,$ solve each equation for the variable.
$$
x^{2}=81
$$

James K.

Numerade Educator

Problem 44

In $43-46,$ solve each equation for the variable.
$$
a^{2}=196
$$

Dilip P.

Numerade Educator

Problem 45

In $43-46,$ solve each equation for the variable.
$$
b^{2}=100
$$

James K.

Numerade Educator

Problem 46

In $43-46,$ solve each equation for the variable.
$$
y^{2}-169=0
$$

Dilip P.

Numerade Educator

Problem 47

The area of a square is 14 square centimeters. What is the length of a side of the square?

James K.

Numerade Educator

Problem 48

Find the length of the hypotenuse of a right triangle if the length of the longer leg is 20 feet and the length of the shorter leg is 12 feet.

Dilip P.

Numerade Educator

Problem 49

Find the length of the shorter leg of a right triangle if the length of the longer leg is 36 inches and the length of the hypotenuse is 39 inches.

James K.

Numerade Educator

Problem 50

What is the length of a side of a square if the length of a diagonal is $\sqrt{72}$ inches?

Dilip P.

Numerade Educator

Problem 51

The length $l,$ width $w,$ and height $h,$ of a rectangular carton are $l=12$ feet, $w=4$ feet, and
$h=3$ feet. The length of a diagonal, $d$ , the distance from one corner of the box to the opposite corner, is given by the formula $d^{2}=l^{2}+w^{2}+h^{2} .$ What is the length of the diagonal of the carton?

James K.

Numerade Educator