5.5 Roots of Real Numbers and Radical Expressions

Definition of n th Root For any real numbers a and b and any positive integers n, if a n = b, then a is the n th root of b. ** For a square root the value of n is 2.

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

Simplify To simplify means to find x in the equation: x 4 = 81 Solution: = 3

Principal Root The nonnegative root of a number Principal square root Opposite of principal square root Both square roots

Summary of Roots even odd one + root one - root one + root no - roots no real roots no + roots one - root one real root, 0

Examples

Taking n th 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 Even Odd

Even Odd Even 2

Product Property of Radicals For any numbers a and b where and,

Product Property of Radicals Examples

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Examples:

Quotient Property of Radicals For any numbers a and b where and,

Examples:

Rationalizing the denominator Rationalizing the denominator means to remove any radicals from the denominator. Ex: Simplify

Simplest Radical Form No perfect nth power factors other than 1. No fractions in the radicand. No radicals in the denominator.

Examples:

Adding radicals We can only combine terms with radicals if we have like radicals Reverse of the Distributive Property

Examples:

Multiplying radicals - Distributive Property

Multiplying radicals - FOIL F O I L

Examples: F O I L

F O I L

Conjugates Binomials of the form where a, b, c, d are rational numbers.

The product of conjugates is a rational number. Therefore, we can rationalize denominator of a fraction by multiplying by its conjugate.

Examples: