1. § 7.5
Multiplying With More Than One Term and Rationalizing Denominators

2. Section objectives In this section, you will learn to:
Multiply radical expressions having more than one term
Use polynomial special products to multiply radicals
Rationalize the denominators containing one term
Rationalize the denominators containing two terms
Blitzer, Intermediate Algebra, 5e – Slide #2 Section 7.5

3. Multiplying Radicals Multiply: Use the distributive property.
EXAMPLE
Multiply:
SOLUTION
Use the distributive property.
Multiply the radicals.
Use FOIL.
Multiply the radicals.
Blitzer, Intermediate Algebra, 5e – Slide #3 Section 7.5

4. Multiplying Radicals
CONTINUED
Factor the third radicand using the greatest perfect square factor.
Factor the third radicand into two radicals.
Simplify.
Blitzer, Intermediate Algebra, 5e – Slide #4 Section 7.5

5. Multiplying Radicals Multiply: Use FOIL. Multiply the radicals.
EXAMPLE
Multiply:
SOLUTION
Use FOIL.
Multiply the radicals.
Group like terms.
Combine radicals.
Blitzer, Intermediate Algebra, 5e – Slide #5 Section 7.5

6. Multiplying Radicals Use the special product for
CONTINUED
Use the special product for
Multiply the radicals.
Blitzer, Intermediate Algebra, 5e – Slide #6 Section 7.5

7. Rationalizing Denominators
EXAMPLE
Rationalize each denominator:
SOLUTION
Using the quotient rule, we can express We
have cube roots, so we want the denominator’s radicand to be a perfect cube. Right now, the denominator’s radicand is We know that If we multiply the numerator and the
denominator of , the denominator becomes
Blitzer, Intermediate Algebra, 5e – Slide #7 Section 7.5

8. Rationalizing Denominators
CONTINUED
The denominator no longer contains a radical. Therefore, we
multiply by 1, choosing
Use the quotient rule and rewrite as the quotient of radicals.
Multiply the numerator and denominator by to remove the radical in the denominator.
Multiply numerators and denominators.
Blitzer, Intermediate Algebra, 5e – Slide #8 Section 7.5

9. Rationalizing Denominators
CONTINUED
Simplify.
(b) The denominator, is a fifth root. So we want the denominator’s radicand to be a perfect fifth power. Right now, the denominator’s radicand is We know that If we multiply the numerator and the denominator
of , the denominator becomes
Blitzer, Intermediate Algebra, 5e – Slide #9 Section 7.5

10. Rationalizing Denominators
CONTINUED
The denominator’s radicand is a perfect 5th power. The denominator no longer contains a radical. Therefore, we
multiply by 1, choosing
Write the denominator’s radicand as an exponential expression.
Multiply the numerator and the denominator by
Multiply the numerators and denominators.
Blitzer, Intermediate Algebra, 5e – Slide #10 Section 7.5

11. Rationalizing Denominators
CONTINUED
Simplify.
Blitzer, Intermediate Algebra, 5e – Slide #11 Section 7.5

12. Rationalizing Denominators
EXAMPLE
Rationalize each denominator:
SOLUTION
The conjugate of the denominator is If we multiply the numerator and the denominator by the simplified denominator will not contain a radical. Therefore, we
multiply by 1, choosing
Multiply by 1.
Blitzer, Intermediate Algebra, 5e – Slide #12 Section 7.5

13. Rationalizing Denominators
CONTINUED
Multiply by 1.
Evaluate the exponents.
Subtract. 3
Divide the numerator and denominator by 4. 1
Blitzer, Intermediate Algebra, 5e – Slide #13 Section 7.5

14. Rationalizing Denominators
CONTINUED
Simplify.
(b) The conjugate of the denominator is If we multiply the numerator and the denominator by the simplified denominator will not contain a radical. Therefore, we
multiply by 1, choosing
Multiply by 1.
Blitzer, Intermediate Algebra, 5e – Slide #14 Section 7.5

15. Rationalizing Denominators
CONTINUED
Multiply by 1.
Rearrange terms in the second numerator.
Simplify.
Blitzer, Intermediate Algebra, 5e – Slide #15 Section 7.5

16. Rationalizing Numerators
EXAMPLE
Rationalize the numerator:
SOLUTION
The conjugate of the numerator is If we multiply the numerator and the denominator by the simplified numerator will not contain a radical. Therefore, we
multiply by 1, choosing
Multiply by 1.
Blitzer, Intermediate Algebra, 5e – Slide #16 Section 7.5

17. Rationalizing Numerators
CONTINUED
Leave the denominator in factored form.
Evaluate the exponents.
Simplify the numerator.
Simplify by dividing the numerator and denominator by 7.
Blitzer, Intermediate Algebra, 5e – Slide #17 Section 7.5

18. GET THOSE RADICALS OUT OF THE DENOMINATOR!!!!
In Summary…
Important to Remember:
Radical expressions that involve the sum and difference of the same two terms are called conjugates. To multiply conjugates, use
The process of rewriting a radical expression as an equivalent expression without any radicals in the denominator is called rationalizing the denominator.
GET THOSE RADICALS OUT OF THE DENOMINATOR!!!!
Blitzer, Intermediate Algebra, 5e – Slide #18 Section 7.5

19. In Summary… On Rationalizing the Denominator…
If the denominator is a single radical term with nth root:
See what expression you would need to multiply by to obtain a perfect nth power in the
denominator. Multiply numerator and denominator by that expression.
If the denominator contains two terms:
Rationalize the denominator by multiplying the numerator and the denominator by
the conjugate of the denominator.
More than two terms in the denominator and rationalizing can get very complicated. Note that you don’t have rules here for those situations. To rationalize simply means to “get the radical out”. By common agreement, we usually rationalize the denominator in a rational expression. We make the denominator “nice” sometimes at the expense of making the numerator messy, but for
comparison and other purposes that you will understand later – this choice is best.
Blitzer, Intermediate Algebra, 5e – Slide #19 Section 7.5