8.5 More Simplifying and Operations with

Radicals

When appropriate, use the rules for adding and multi-
plying polynomials to simplify radical expressions.

The formulas

and

are useful when simplifying radical expressions.

1 ab 21 ab 2 a^2 b^2

1 ab 22 a^2  2 abb^2 ,

1 ab 22 a^2  2 abb^2 ,

Distributive property

FOIL

Combine like terms.

Multiply fractions.

= Subtract.

6 A 27 + 22 B

5

=

6 A 27 + 22 B

7 - 2

=

(^6) A 27 + (^22) B
A^27 -^22 BA^27 +^22 B
6
27 - 22
3
26

3 # 26
26 # 26

326
6

26
2
= 5 - 3, or 2
A^25 +^23 BA^25 -^23 B
= 15 - 2226
= 13 - 2226 + 2
=A 213 B^2 - 2 A 213 BA 22 B +A 22 B^2
A^213 -^22 B
2
= 1 - 23
= 3 - 223 + 23 - 2
A 23 + 1 BA 23 - 2 B
26 A 25 - 27 B = 230 - 242
CONCEPTS EXAMPLES
546 CHAPTER 8 Roots and Radicals
Any denominators with radicals should be rationalized.
If a radical expression contains two terms in the denomi-
nator and at least one of those terms is a square root
radical, multiply both numerator and denominator by the
conjugate of the denominator.
Multiply by the
conjugate of the
denominator.
8.6 Solving Equations with Radicals
Solving a Radical Equation
Step 1 Isolate a radical.
Step 2 Square each side. (By the squaring property
of equality, all solutions of the original
equation are amongthe solutions of the squared
equation.)
Step 3 Combine like terms.
Step 4 If there is still a term with a radical, repeat
Steps 1–3.
Step 5 Solve the equation for proposed solutions.
Step 6 Check all proposed solutions from Step 5 in the
original equation.
Solve
Isolate the radical.
Square each side.
Standard form
Factor.
Zero-factor property
Proposed solutions
Verify that 2 is the only solution (6 is extraneous).
The solution set is 526.
x=2or x= 6
x- 2 =0orx- 6 = 0
0 = 1 x- 221 x- 62
0 =x^2 - 8 x+ 12
2 x- 3 = 9 - 6 x+x^2
A 22 x- 3 B
2
= 13 - x 22
22 x- 3 = 3 - x
22 x- 3 +x=3.
(continued)
Remember the middle
term when squaring the
binomial on the right.
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