5.5 Roots of Real Numbers and Radical

Expressions

Definition of nth Root

** For a square root the value of n is 2.

For any real numbers a and b and any positive integers n, if an = b, then a is the nth root of b.

Definitions

A perfect square is the square of a natural number. 1, 4, 9, 16, 25, and 36 are the first six perfect squares.

A perfect cube is the cube of a natural number. 1, 8, 27, 64, 125, and 216 are the first six perfect cubes.

Notation

814index

radical

radicand

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

Simplifying Radicals

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

yEven

Even

3 x

Odd

Odd

Even

Even

2

323- y

323- y