Math 95 Lecture Notes

Chapter 7: Radical Expressions and Functions

Spiral of Theodorus

Square Roots

• The number b is a square root of a if b2 = a.

• The principal square root is the positive square root.

• The square root of a negative number is NOT a real number.

Example 1. List the two distinct square roots of the number 9.

Example 2. Evaluate the following expressions. Simplify each as much as possible.

(a)√

36 (b)√

425

(c)√

0.81 (d)√a2, a > 0

1

Math 95 Lecture Notes Chapter 7: Radical Expressions and Functions

Cube Roots

• The number b is a cube root of a if b3 = a.

• The cube root of a negative number is a real number.

Example 3. Evaluate the following expressions. Simplify each as much as possible.

(a) 3√

64 (b) 3√−8 (c)

3√a6 (d) 3

√127

nth Roots

• The number b is an nth root of a if bn = a.

• An even root of a negative number is NOT a real number.

• Ann odd root of a negative number is a real number.

Example 4. Evaluate the following expressions. Simplify each as much as possible.

(a) 4√

81 (b) 4√−81 (c) 5

√32 (d) 5

√−32

Example 5. The period T (in seconds) of a pendulum as a function

of its length L (in feet) is given by T = 2π

√L

32.2. A clock in London

(known as “Big Ben”) has a pendulum length of 13 feet. What is the

period of this pendulum?

Instructor: A.E.Cary Page 2 of 6

Math 95 Lecture Notes Chapter 7: Radical Expressions and Functions

Product and Quotient Rules for Radical Expressions

Let a and b be real numbers where both n√a and n

√b are defined.

• n√a · n√b = n√a · b

•n√a

n√b

= n√

ab, b 6= 0

Example 6. Simplify each expression. Assume all variables are positive.

(a)√

60

(b)√

3 ·√

27

(c)√5010

(d) 7√24

(e)√

5 ·√

5

(f)√

6 ·√

8

(g) 1√3

(h) 5√30

(i) 3√−4 · 3

√−4 · 3

√−4

(j)√x ·√x5

(k)√3√5

(l)√2√8

Instructor: A.E.Cary Page 3 of 6

Math 95 Lecture Notes Chapter 7: Radical Expressions and Functions

Example 7. Create a table of function values for f(x) =√x. Then use these values to sketch the

graph of y = f(x) onto Figure 1.

Figure 1

Domain of f :

Range of f :

Example 8. Create a table of function values for f(x) = 3√x. Then use these values to sketch the

graph of y = f(x) onto Figure 2.

Figure 2

Domain of f :

Range of f :

Instructor: A.E.Cary Page 4 of 6

Math 95 Lecture Notes Chapter 7: Radical Expressions and Functions

Example 9. If possible, find f(−2) and f(3) for the following functions. Simplify each as much as

possible.

(a) f(x) =√

10− x

(b) f(x) = 3√x− 6

(c) f(x) =√x3 − 2

Example 10. State the domain of each of the functions below using set-builder notation.

(a) f(x) =√

3− x+ 5

(b) g(x) =√x2 + 4

(c) h(x) = 3√x− 5

(d) k(x) = 5√2x+6

Instructor: A.E.Cary Page 5 of 6

Math 95 Lecture Notes Chapter 7: Radical Expressions and Functions

Example 11. Graph y = f(x) for the following functions using your graphing calculator. It will be

helpful to find the domain of each function first. After graphing, clearly state the the domain and range

of each function.

(a) f(x) =√

2x+ 3− 4

Figure 3

(b) f(x) = −2√x+ 7 + 3

Figure 4

Instructor: A.E.Cary Page 6 of 6