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2. To simplify radicals, we will use the laws of radicals. Radicals are
like exponents. Study how the laws of radicals are obtained from the
laws of exponents.
Law of Radicals Proof
1.
𝑛
𝑥𝑚 = x 𝑛
𝑥𝑚 = 𝑥
𝑚
𝑛 = x
2. 𝑛
𝑥𝑦 = 𝑛
𝑥 . 𝑛
𝑦
Product Property of Radicals
𝑛
𝑥𝑦 = (𝑥𝑦)
1
𝑛 = 𝑥
1
𝑛 𝑦
1
𝑛 = 𝑛
𝑥 𝑛
𝑦
3.
𝑛 𝑥
𝑦
=
𝑛
𝑥
𝑛 𝑦
𝑛 𝑥
𝑦
=
𝑥
𝑦
1
𝑛
=
𝑥
1
𝑛
𝑦
1
𝑛
=
𝑛
𝑥
𝑛 𝑦

3. Example 1. Simplify
3
24.
Steps Solution
1. Factor the radicand such that one of
the factors is a perfect cube.
8 is a factor of 24 which is a perfect
cube,
3
24 =
3
8 =
3
3
2. Use the product property of radicals. 3
24 =
3
8 .
3
3
3. Simplify. =
3
8 .
3
3
=2 .
3
3
= 2
𝟑
𝟑

4. Example 2. Simplify 200𝑡3𝑢5.
Steps Solution
1. Find the perfect square factor in the
radicand.
200𝑡3𝑢5 = 100𝑡2𝑢4 . 2𝑡𝑢
2. Use the product property of radicals
and simplify.
100𝑡2𝑢4 2𝑡𝑢
= 10𝑡
2
2𝑢
4
2 2𝑡𝑢
= 10tu2 2𝑡𝑢

5. Example 3. Simplify
4
32𝑤9𝑥7.
Steps Solution
1. Factor the radicand such that one of
the factors has fourth root which is an
integer.
16 is a factor of 32. Also, 16 has a perfect
fourth root because 16 = 24 .
4
32𝑤9𝑥7 =
4
16. 2 . 𝑤9 . 𝑥7
2. Use the product property of radicals. 4
32𝑤9𝑥7 =
4
16 . 𝑤8 . 𝑥4 4
2 . 𝑤1 . 𝑥3
= 2w2x
𝟒
𝟐𝒘𝒙𝟑

6. Thank
You