1. Multiplying and Dividing Radical Expressions
Section 9.4
Multiplying and Dividing Radical Expressions

2. Objectives Multiply radical expressions
Find powers of radical expressions
Rationalize denominators
Rationalize denominators that have two terms
Rational numerators

3. Objective 1: Multiply Radical Expressions
We have used the product rule for radicals to write radical expressions in simplified form. We can also use this rule to multiply radical expressions that have the same index.
The Product Rule for Radicals:
The product of the nth roots of two nonnegative numbers is equal to the nth root of the product of those numbers.

4. EXAMPLE 1
Multiply and then simplify, if possible:
Strategy In each expression, we will use the product rule for radicals to multiply factors of the form .
Why The product rule for radicals is used to multiply radicals that have the same index.

5. EXAMPLE 1
Multiply and then simplify, if possible:
Solution

6. EXAMPLE 1
Multiply and then simplify, if possible:
Solution
b. We use the commutative and associative properties of multiplication to multiply the integer factors and the radicals separately. Then we simplify any radicals in the product, if possible.

7. EXAMPLE 1
Multiply and then simplify, if possible:
Solution

8. Objective 2: Find Powers of Radical Expressions
The nth Power of the nth Root: If is a real number,

9. EXAMPLE 4
Strategy In part (a), we will use the definition of square root. In part (b), we will use a power rule for exponents. In part (c) and (d), we will use the FOIL method.
Why Part (a) is the square of a square root, part (b) has the form , and part (c) has the form

10. EXAMPLE 4
Solution a.
b. We can use the power of a product rule for exponents to find

11. EXAMPLE 4
Solution

12. EXAMPLE 4
Solution
d. We can use the FOIL method to find the product.

13. Objective 3: Rationalize Denominators
Simplified Form of a Radical Expression:
1. Each factor in the radicand is to a power that is less than the index of the radical.
2. The radicand contains no fractions or negative numbers.
3. No radicals appear in the denominator of a fraction.
We now consider radical expressions that do not satisfy requirements 2 or 3. We will introduce an algebraic technique, called rationalizing the denominator, that is used to write such expressions in an equivalent simplified form. In this process, we multiply the expression by a form of 1 and use the fact that

14. Objective 3: Rationalize Denominators
As an example, let’s consider the following expression:
This radical expression is not in simplified form, because a radical appears in the denominator. It doesn’t satisfy requirement 3 listed above.
These equivalent fractions represent the same number, but have different forms. Since there is no radical in the denominator, and is in simplest form, the expression is in simplified form.

15. EXAMPLE 5
Rationalize the denominator:
Strategy We look at each denominator and ask, “By what must we multiply it to obtain a rational number?” Then we will multiply each expression by a carefully chosen form of 1.
Why We want to produce an equivalent expression that does not have a radical in its denominator.

16. EXAMPLE 5 Solution Rationalize the denominator:
a. This radical expression is not in simplified form, because the radicand contains a fraction. We begin by writing the square root of the quotient as the quotient of two square roots:
To rationalize the denominator, we proceed as follows:
The denominator is now a rational number, 7.

17. EXAMPLE 5 Solution Rationalize the denominator:
b. This expression is not in simplified form because a radical appears in the denominator of a fraction. Here, we must rationalize a denominator that is a cube root. We multiply the numerator and the denominator by a number that will give a perfect cube under the radical.

18. Objective 4: Rationalize Denominators That Have Two Terms
To rationalize the denominator of a two term denominator such as , we multiply the numerator and denominator by , because the product contains no radicals.
Radical expressions that involve the sum and difference of the same two terms, such as , are called conjugates.

19. EXAMPLE 9
Rationalize the denominator:
Strategy In each part, we will rationalize the denominator by multiplying the numerator and the denominator by the conjugate of the denominator.
Why Multiplying each denominator by its conjugate will produce a new denominator that does not contain radicals.

20. EXAMPLE 9
Rationalize the denominator:
Solution

21. EXAMPLE 9
Rationalize the denominator:
Solution

22. Objective 5: Rationalize Numerators.
In calculus, we sometimes have to rationalize a numerator by multiplying the numerator and denominator of the fraction by the conjugate of the numerator.

23. EXAMPLE 10
Rationalize the numerator:
Strategy To rationalize the numerator, we will multiply the numerator and the denominator by the conjugate of the numerator.
Why After rationalizing the numerator, we can simplify the expression. Although the result will not be in simplified form, this non simplified form is often desirable in calculus.

24. EXAMPLE 10
Rationalize the numerator:
Solution
We multiply the numerator and denominator by , which is the
conjugate of the numerator.