Roots of Real Numbers and Radical

Expressions

Definition of Square Root

For any real numbers a and b, if , then a is a square root of b.Ex: , 5 is a square root of 25

ba 2

2552

Definition of nth Root• For an real numbers a and b, and any positive n,

if , then a is an nth root of b.

• EX: Since , 2 is a fifth root of 32.

ban

3225

Notation

814indexradical

radicand

Note: An index of 2 is understood but not written in a square root sign.

Simplify 814

To simplify means to find x in the equation:

x4 = 81

Solution: = 3 814

Principal Root

The nonnegative root of a number

64 64 64

Principal square root

Opposite of principal square root Both square roots

Summary of Roots

even

odd

one + root one - rootone + root no - roots

no real roots

no + roots one - root

one real root, 0

Examples

4

4

1. 169

2. - 8 -3

x

x

2213 x 213x

228 3 x

28 3x

Examples

323 5 x3 6

3 3 3

3. 125

4.

x

m n

25 x

33 mn mn

Taking nth roots of variable expressions

Using absolute value signs

If the index (n) of the radical is even, the power under the radical sign is even, and the resulting power is odd, then we must use an absolute value sign.

Examples

44

626

1.

2.

an

xy

EvenEven

EvenEven

an

Odd

2 x yOdd

6

1826

3.

4. 3

x

yEvenEven

3 x

Odd

Odd

Even

Even

2

323- y

323- y