To divide radicals, we use the Division Property of Radicals which states that n√x/√y = n√x/√y if the radicals have the same index. Examples are provided to demonstrate dividing radicals with the same index and rationalizing radicals in the denominator by multiplying the numerator and denominator by the radical.

2. To divide radicals, we use the Division Property of Radicals.
𝑛 𝑥
𝑦
=
𝑛
𝑥
𝑛 𝑦
Remember that radicals can be divided if they have the same
index.

3. Example 1. (12 𝟏𝟖) ÷ (2 𝟖). Example 2. Divide
𝟐𝟓
𝟏𝟔
.
= (12 ÷ 2) (18 ÷ 8)
= 6
18
8
(2) = 6
9
4
= (6)
9
4
= 6
3
2
=
18
2
= 9
=
25
16
=
𝟓
𝟒

4. Example 3.
𝟏𝟖
𝟑
𝟓𝟒𝒎𝟕
𝒏𝟕
𝟐
𝟑
𝟒𝒎𝒏
Example 4. Simplify
𝟒𝟓𝒂𝟓
𝟗𝒂𝟐
= 9
3
54𝑚7
𝑛7
3
2𝑚𝑛
= 9
3
27𝑚6𝑛6
= 9 (3m2n2)
= 27m2n2
=
45𝑎5
9𝑎2
= 5𝑎3
= 𝑎2 . 5 . 𝑎1
= a 𝟓𝒂

5. Example 5. ( 3) ÷
4
3).
= ( 3) ÷
4
3)
= 3
1
2 ÷ 3
1
4
= 3
1
2 -
1
4
= 𝟑
𝟏
𝟒 or
𝟒
𝟑

6. Example 6. Divide and simplify
5
3
.
Notice that we can no longer use the division property of
radicals to further simplify
5
3
. However, a radical in the
denominator of a fraction is not allowed. To remove a radical from
the denominator, we are going to rationalize the denominator.

7. Steps Solution
1 Rationalize the denominator by
multiplying both numerator and
denominator of the fraction by 3.
=
5
3
=
5
3
.
3
3
2. Use the product property of radicals
and simplify.
=
15
9
=
𝟏𝟓
𝟑

8. Example 7. Divide and simplify
5
2
.
Steps Solution
1. Rationalize the denominator. =
5
2
.
2
2
2. Use the product property of radicals
and simplify.
=
10
4
=
10
2