Math 152 — Rodriguez
Blitzer — 7.1
Radical Expressions and Functions
I. Square Roots
A. Definition: A square root of a is b if b2 = a.
A square root of 9 is 3 since 32 = 9.
Another square root of 9 is –3 since (–3)2 = 9.
Example:
B. To denote the nonnegative square root of a nonnegative number we write
called the principal square root. The number a is called the radicand.
a . This is
C. If we want the negative square root of a number we write it as − a .
D. Evaluate or state that the expression is not a real number.
1.
0
6.
0.49
2.
81
7.
1
4
3.
25
8.
144 + 25
9.
70
4. − 25
5.
−25
Observations:
10.
40
positive perfect square =
positive but not a perfect square =
negative number =
E. Since we defined x to represent the nonnegative root of the number, we must make
sure that ‘what comes out’ of the square root is nonnegative.
Examples: Simplify.
1.
62
2.
( −6 )
3.
x2
4.
2
x4
5.
(x−4)
6.
( x +5)
2
2
F. The square root function is defined by f(x) = x .
()
Example: Let f x = x − 2 . Find the indicated function values.
Find f(3), f(2), f(1), and f(0).
G.
Based on the above example, what can we say about the domain of a square root
function?
Examples:
()
()
1) Find the domain of f x = x − 2 .
2) Find the domain of f x = 4 − x .
II. Cube Roots
A. Definition: The cube root of a is b if b3 = a.
Example: The cube root of 8 is 2 since 23 = 8. It is the ONLY cube root of 8.
The cube root of –8 is –2 since (–2)3 = –8. It is the ONLY cube root of –8.
B. The cube root is denoted by 3 x . The number x is the radicand. The number 3 is the
index. The index tells us what root we are ‘taking’.
Examples: Find each cube root.
1. 3 −27
4.
3
−125
2.
3
1
5. − 3 64
3.
3
0
6.
3
1
8
()
C. The cube root function is defined by f x = 3 x .
Example: Find the indicated function values.
()
f x = 3x−2.
Find f(10), f(1), f(2), and f(–6).
D. Based on the above example, what is the domain of a cube root?
Blitzer — 7.1
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III. Higher Roots
A. The nth root of a is b if bn = a. It is denoted by n a . The number a is the radicand. The
number n is the index, which is again, the root we are taking.
B. A cube root has an index of 3. A square root has an index of 2, which is not written.
C. If the index is an odd number, we say we are taking an odd root. All odd roots have
the same characteristics of a cube root.
a. Every (real) number has only ONE odd root.
b. An odd root of a negative number is negative.
c. Ad odd root of a positive number is positive.
d.
n
an = a .
D. If the index is an even number, we say we are taking an even root. All even roots
have the same characteristics of a square root.
a. Every positive number has two roots—one positive and one negative.
b. We use
n
a to denote the principal nth root; that is the nonnegative root of a.
c. We use − n a to denote the negative root of the number.
d. An even root of a negative number is not a real number.
e.
n
an = a .
Examples: Simplify. Include absolute value bars where necessary.
1.
4
16
6.
2.
4
−81
7. − 5 32
3. − 4 81
8.
4.
5
−1
9.
5.
5
32
IV. Application
5
3
10.
−32
−27x 3
4
16x 4
4
( x − 2)
4
()
Police use the function f x = 20x to estimate the speed of a car, f(x), in miles per hour,
based on the length, x, in feet, of it skid marks up on sudden braking on a dry asphalt road. A
motorist is involved in an accident. A police officer measures the car’s skid marks to be 45
feet long. Estimate the speed at which the motorist was traveling before braking.
Blitzer — 7.1
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