Section 10.5

Expressions Containing Several Radical Terms

Definition

Like Radicals are radicals that have the same index and same radicand.

We can ONLY combine Like Radicals.

• 1) Simplify each radical.• 2) Combine like radicals.

To add/subtract radical expressions, we

Solution

Simplify by combining like radical terms.

3 3 32 2 2

a) 3 5 7 5

b) 7 9 9 2 9s s s

a) 3 5 7 5 (3 7) 5= 10 5

3 3 3 32 2 2 2b) 7 9 9 2 9 (7 1 2) 9s s s s

3 26 9s

Example

Solution

Simplify by combining like radical terms.

3 43

a) 2 18 7 2

b) 10m 3 24m m

a) 2 18 7 2 2 9 2 7 2

2(3) 2 7 2 6 2 7 2 2

3 43 3 3b) 10m 3 24 10m 3 2 3m m m m m 312 3m m

Example

ExamplesSimplify the following expressions

aaa 735

482122753

33 4 24 xxx 333 185032 xxx

254 3 xx

Product of two or more radical terms1. Use distributive law or FOIL

2. Use product rule for radicals

3. Simplify and combine like terms.

Examples: Multiply. Simplify if possible. Assume all variables are positive

nnn baab

xxn n

3 23

a) 2( 2)

b) 2 3

c)

y

x x

m n m n

Solution

a) 2( 2) 2 2 2y y

2 4 2 2y y

Using the distributive law

3 3 32 2 23 3 3b) 2 3 3 2 6x x x x x x

3 33 233 2 6x x x

3 233 2 6x x x

F O I L

Solution

c) m n m n

2 2m m n m n n

m n

F O I L

Notice that the two middle terms are opposites, and the result contains no radical. Pairs of radical terms like,

are called conjugate pairs.

and ,m n m n

Rationalizing Denominators with Two

Terms The sum and difference of the same terms are

called conjugate pairs.

To rationalize denominators with two terms, we multiply the numerator and denominator by the conjugate of the denominator.

Solution

Rationalize the denominator:3

5 2

5 2

5 2

3( 5 2)

25 10 10 4

3 3

5 2 5 2

Example

3( 5 2)

5 2

3( 5 2)

3

5 2

1

1

Solution

Rationalize the denominator:5

.7 y

7

7

y

y

2

5 7 5( 7 )

7 7 49 7 7

y y

y y y y y

25 7 5

7

y

y

5 5.

7 7y y

Example

Solution

Rationalize the denominator:4 m

m n

m n

m n

2

2 2

4 4m mn

m mn mn n

4 4.

m m

m n m n

Example

4 4m mn

m n

Terms with Differing Indices

To multiply or divide radical terms with different indices, we can convert to exponential notation, use the rules for exponents, and then convert back to radical notation.

Solution

Multiply and, if possible, simplify: 5 3 .x x

5 3 1/ 2 3/ 5x x x x 11/10x

10 11x

10 10 10x x 10x x

Converting to exponential notation

Adding exponents

Converting to radical notation

Simplifying

Example

Group ExerciseSimplify the following radical expressions

2( 3 2)x 6

7 4

24

5

2

a

a

233 32 3 4y y y