6.3 Simplifying Radical Expressions

In this section, we assume that all variables are positive.

The Product Rule for Radicals

For any nonnegative real numbers a and b and any index k,

(To multiply, multiply the radicands.)

Multiply and simplify radical expressions.

5) Multiply and simplify radical expressions.

For any nonnegative real numbers a and b and any index k,

(Take the kth root of each factor separately.)

Factoring Radical Expressions

Simplifying kth Roots

To simplify a radical expression by factoring:1. Look for the largest factors of the radicand that are perfect kth powers (where k is the index).2. Then take the kth root of the resulting factors.3. A radical expression, with index k, is simplified when its radicand has no factors that are perfect kth powers.

Multiply and simplify radical expressions.

9) Simplify by factoring.

10) Multiply and simplify radical expressions.

11) Multiply and simplify radical expressions.

12) Multiply and simplify radical expressions.

The Quotient Rule for Radicals

For any nonnegative number a, any positive number b, and any index k,

(To divide, divide the radicands. After doing this, you can sometimes simplify by taking roots.)

13) Divide and simplify

=

14) Divide and simplify radical expressions.

For any nonnegative number a, any positive number b, and any index k,

(Take the kth roots of the numerator and of the denominator separately.)

15) Simplify

16) Divide and simplify radical expressions.

17) Divide and simplify radical expressions.

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ba

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ba