Download Unit 6 Lesson 2 Operations on Radicals Addition and Subtraction

Unit 6 Lesson 2
Operations on Radicals
Addition and Subtraction
 Adding and Subtracting radicals is similar to adding and subtracting
polynomials.
 Just as you cannot combine 3x and 6y, since they are not like terms, you
cannot combine radicals unless they are like radicals.
 If asked to simplify the expression 2x + 3x, we recognize that they each share
a common variable part x that makes them like terms, hence we add the
coefficients and keep the variable part the same.
3x + 2x = 5x
 Likewise, if asked to simplify radicals. If they have “like” radicals then we
add or subtract the coefficients and keep the like radical. Two radical
expressions are said to be like radicals if they have the same index and
the same radicand.

2 5 +3 5 = 5 5
If the radicals in your problem are different, be sure to check to see if the
radicals can be simplified. Often times, when the radicals are simplified,
they become the same radical and can then be added or subtracted. Always
simplify first, if possible.
4 5 + 3 125
4 5 + 3× 5 5
4 5 +15 5
19 5
Example 1: Simplifying Radicals
Step 1: Divide the number under the radical. If all numbers are not prime, continute
dividing.
Step 2: Find pairs, for a square root, under the radical and pull them out.
Step 3: Multiply the items you pulled out by anything in front of the radical sign.
Step 4: Multiply anything left under the radical.
-8 99x 5 y 3 z 2
-8 9 ×11× x × x × x × x × x × y × y × y × z × z
-8 3× 3×11× x × x × x × x × x × y × y × y × z × z
-8× 3× x × x × y × z 11xy
-24x 2 yz 11xy
Example 2: Simplify
3 + 12
Unit 6 Lesson 2
Operations on Radicals
Example 3: Simplify
6 +2 6
Example 4: Simplify 4 7 +8 63
DO NOT ADD THE NUMBERS UNDER THE RADICAL!
You may not always be able to simplify radicals.
Since the radicals are not the same, and both are in their simplest form, there is no
way to combine them. The answer is the same as the problem:
Example 5: Simplify 4 7 + 2 3
Example 6: Simplify 2 4 2x + 6 4 2x
Example 7: Simplify 4 3 3x + 5 3 10x
Example 8: A garden has a width of 13 and a length of 7 13 . What is the
perimeter of the garden in simplest radical form?
Unit 6 Lesson 2
Operations on Radicals
Muliplication
To multiply radicals, consider the following:
9 × 4 = 3× 2 = 6 and 9× 4 = 36 = 6
3
-8 × 3 64 = -2 × 4 = -8 and 3 -8× 64 = 3 -512 = -8

Multiplying Radical Expressions
n

a × n b = n ab
You can simplify the product of two radicals by writing them as one radical
them simplifying.
Here is an example of how you can use this Multiplication Property to
simplify a radical expression.
Example 1:
3
7x 7 × 3 6x8
Example 2: 15x 3 y × 15xy
Example 3: 7x( x - 7 7)
 Recall that to multiply polynomials you need to use the Distributive Property.
(a + b)(c – d)
a(c – d) + b(c – d)
ac – ad + bc – bd
Unit 6 Lesson 2
Operations on Radicals
Conjugates
Example 1:

6( 6 - 2)+ 2( 6 - 2)
Notice that just like a difference of two squares, the middle terms cancel out.
 We call these conjugate pairs.
The conjugate of a + b is a - b

They are a conjugate pair. When we multiply a conjugate pair, the radical
cancels out and we obtain a rational number.

Simplified Radical Expressions are recognized by…
No radicands have perfect square factors other than 1.
 No radicands contain fractions.
 No radicals appear in the denominator of a fraction.
Unit 6 Lesson 2
Operations on Radicals
Dividing Radicals
To divide radicals, consider the followng:
16 4
= = 2 and
4 2
3
64 4
= = 2 and
3
8 2
16
16
=
= 4 =2
4
4
3
64 3 64 3
=
= 8=2
3
8
8

Dividing Radical Expressions
n
a na
=
n
b
b

You can simplify the quotient of two radicals by writing them as one radical
then simplifying.
Example 1:
90x18
2x
Example 2:
6x 8 y 9
5x 2 y 4



Notice that after simplifying the radical, we still have a square root in the
denominator.
We have to find a way to get rid of the radical.
This is called rationalizing the denominator.
Unit 6 Lesson 2
Operations on Radicals
Rationalizing the Denominator
Case 1: There is ONE TERM in the denominator and it is a SQUARE ROOT.
When the denominator is a monomial (one term), multiply both the numerator and
the denominator by whatever makes the denominator an expression that can be
simplified so that it no longer contains a radical. In this case it happens to be exactly
the same as the denominator.
7
2
7 2
14
14
×
=
=
2
2 2
4
Case 2: There is ONE TERM in the denominator, however, THE INDEX IS GREATER
THAN TWO.
Sometimes you need to multiply by whatever makes the denominator a perfect cube
or any other power greater than 2 that can be simplified.
3
9
3
11
9 3 11×11 3 1089
=
=
3
11
11 3 11×11
3
Unit 6 Lesson 2
Operations on Radicals
Case 3: There are TWO TERMS in the denominator. We also use conjugate pairs to
rationalize denominators.
Be sure to enclose expressions with multiple terms in ( ). This will help you to
remember to FOIL these expressions. Always reduce the root index (numbers
outside radical) to the simplest form (lowest) for the final answer.
3- 6
3+ 6
=
3- 6
3- 6
·
3+ 3
3- 6
=
9 - 18 - 18 - 36
9 - 18 + 18 - 36
=
3- 2 18 - 6
9 - 36
-3 - 2(3 3)
3- 6
-3 - 6 3
=
-3
= 1+ 2 3
=