11-2: OPERATIONS WITH RADICAL EXPRESSIONSEssential Question: How do you add and subtract radicals?

11-2: Operations With Radical Expressions

Like Radicals: Radicals that have the same radicand (term underneath the root)

Like radicals, similar to like terms, can be added and subtracted.

Example 1: Combining Like Radicals Simplify

Both terms contain , so they can be combined If there is no number in front of a radical, assume

a “1”

2 3 22

1 2 3 2 4 2

11-2: Operations With Radical Expressions

Your Turn: Simplify each expression

3 5 4 5

10 5 10

7 5

4 10

11-2: Operations With Radical Expressions

Sometimes, radicals that don’t appear to be like terms can be combined if you simplify all radicals.

Example 2: Simplifying to Combine Like Radicals Simplify

12 can be broken down: 2 ● 2 ● 3 =

7 3 122 3

7 3 2 3 5 3

11-2: Operations With Radical Expressions

Your Turn: Simplify each expression

3 20 2 5

3 3 2 27

8 5

3 3

11-2: Operations With Radical Expressions

When simplifying an expression like , use the distributive property to

multiply the outside term with all inside terms. Remember: Numbers under roots can

multiply with numbers under roots only. Simplify

18 can be simplified (3 ● 3 ● 2)

3 6 7

3 6 718 7 3

3 2 7 3

11-2: Operations With Radical Expressions

Your Turn: Simplify each expression

5 2 10

2 6 11x x

2 5 5 2

2 3 11 2x x

11-2: Operations With Radical Expressions

If both radicals have two terms, you can multiply the same way you do two binomials – by using FOIL

Example 4: Using FOIL Simplify

FIRST: OUTER: INNER: LAST:

Put it all together:

2 15 15 5 5 5 55

75 15 5 5 3 2515 5 10 32 7

152 5 2 01 2 2 5 3

35 05 210 3 5 33 5

11-2: Operations With Radical Expressions

YOUR TURN: Simplify each expression

2 6 3 3 6 5 3

27 4

33 21 2

23 8 7

11-2: Operations With Radical Expressions

Conjugates: The sum and difference of the same two terms. The product of two conjugates results in the difference of two squares.

Notice that the product of conjugates no longer has a radical.

This becomes important when rationalizing a denominator that has two terms

105 2 5 2 25 40

5 2

3

1

11-2: Operations With Radical Expressions

Example 5: Rationalizing a Denominator Using Conjugates Simplify

Need to get rid of square roots in denominator, so multiply top and bottom of the fraction by the conjugate of the denominator. Top is distribution (or FOIL), bottom is FOILed

6

5 2

5 2

5 2 10 10

6 6 5 6 2

5 2 25 4

6 5 6 2

5 2

6 5 6 22 5 2 2

3

Divide all #soutside by 3

11-2: Operations With Radical Expressions

YOUR TURN: Simplify each expression

4

10 8

2 7 2 5

2 10 4 2

4

7 5

11-2: Operations With Radical Expressions

Assignment Worksheet #11-2 Problems 1 – 45, odds