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Radicals and Rational Exponents
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2. 9.5 Multiplying Radical Expressions and Rationalizing Section
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3. Objectives
Multiply two radical expressions. Rationalize the denominator of a fraction that contains a radical expression. Rationalize the numerator of a fraction that contains a radical expression. Solve an application containing a radical expression. 1 2 3 4

4. Multiply two radical expressions1.

5. Multiply two radical expressions
Radical expressions with the same index can be multiplied and divided.

6. Example 1
Multiply: a. b. Solution: We use the commutative and associative properties of multiplication to multiply the coefficients and the radicals separately. Then we simplify any radicals in the product, if possible. a.
Multiply the coefficients and
multiply the radicals.
3(2) = 6 and

7. Example 1 – Solution
cont’d
Simplify.

8. Example 1 – Solution
cont’d b.
Multiply the radicals.
Factor 27a5.

9. Multiply two radical expressions
To multiply a radical expression with two or more terms by a radical expression, we use the distributive property to remove parentheses and then simplify each resulting term, if possible. To multiply two radical expressions, each with two or more terms, we use the distributive property as we did when we multiplied two polynomials. Then we simplify each resulting term, if possible.

10. Multiply two radical expressions
Comment It is important to draw radical signs so they completely cover the radicand, but no more than the radicand. For example and are not the same expressions. To avoid confusion, we can use the commutative property of multiplication and write an expression such as in the form

11. Rationalize the denominator of a fraction that contains a radical expression2.

12. Rationalize the denominator of a fraction that contains a radical expression
To divide radical expressions, we rationalize the denominator of a fraction to write the denominator with a rational number. For example, to divide by we write the division as the fraction

13. Rationalize the denominator of a fraction that contains a radical expression
To rationalize the radical in the denominator, we multiply the numerator and the denominator by a number that will result in a perfect square under the radical. Because 3  3 = 9 and 9 is a perfect square, is such a number.
Multiply numerator and denominator by
Multiply the radicals.

14. Rationalize the denominator of a fraction that contains a radical expression
Since there is no radical in the denominator and cannot be simplified, the expression is in simplest form, and the division is complete.

15. Example 5
Rationalize each denominator. a. b. Solution: a. We first write the square root of the quotient as the quotient of two square roots.

16. Example 5 – Solution
cont’d
Because the denominator is a square root, we must then multiply the numerator and the denominator by a number that will result in a rational number in the denominator. Such a number is
Multiply numerator and denominator by
Multiply the radicals.
Simplify

17. Example 5 – Solution
cont’d
b. Since the denominator is a cube root, we multiply the numerator and the denominator by a cube root of a number that will result in a perfect cube under the radical sign. Since 8 is a perfect cube, and 8 = 2  4, then is such a number.
Multiply numerator and denominator by
Multiply the radicals in the denominator.

18. Example 5 – Solution
cont’d
Simplify.

19. Rationalize the denominator of a fraction that contains a radical expression
To rationalize the denominator of a fraction with square roots in a binomial denominator, we can multiply the numerator and denominator by the conjugate of the denominator. Conjugates The conjugate of (a + b) is (a – b), and the conjugate of (a – b) is (a + b).

20. Rationalize the denominator of a fraction that contains a radical expression
If we multiply an expression such as by its conjugate we will obtain an expression without any radical terms.

21. Rationalize the numerator of a fraction that contains a radical expression3.

22. Rationalize the numerator of a fraction that contains a radical expression
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: Solution: We multiply the numerator and denominator by which is the conjugate of the numerator.

24. Solve an application containing a radical expression4.

25. Example 11 – Photography
Many camera lenses (see Figure 9-15) have an adjustable opening called the aperture, which controls the amount of light passing through the lens.
Figure 9-15

26. Example 11 – Photography
The f-number of a lens is its focal length divided by the diameter of its circular aperture. f-number = A lens with a focal length of 12 centimeters and an aperture with a diameter of 6 centimeters has an f-number of and is an f/2 lens. If the area of the aperture is reduced to admit half as much light, the f-number of the lens will change. Find the new f-number.
f is the focal length, and d is the diameter of the aperture.

27. Example 11 – Solution
We first find the area of the aperture when its diameter is 6 centimeters. A =  r2 A =  (3)2 A = 9 When the size of the aperture is reduced to admit half as much light, the area of the aperture will be square centimeters.
The formula for the area of a circle.
Since a radius is half the diameter, substitute 3 for r.

28. Example 11 – Solution
cont’d
To find the diameter of a circle with this area, we proceed as follows:
A =  r2
This is the formula for the area of a circle.
Substitute for A and for r.
Multiply both sides by
the reciprocal of the coefficient of d2
Simplify.

29. Example 11 – Solution
cont’d
Since the focal length of the lens is still 12 centimeters and the diameter is now centimeters, the new f-number of the lens is
Substitute 12 for f and for d.
Simplify.
Rationalize the denominator.

30. Example 11 – Solution  2.828427125 The lens is now an f/2.8 lens.
cont’d
 The lens is now an f/2.8 lens.
Simplify.
Use a calculator.