To simplify radicals, we use the laws of radicals which are derived from the laws of exponents. Specifically, the product property of radicals states that the radical of a product is equal to the product of the radicals, allowing us to simplify radicals that have perfect nth root factors in their radicands. This property is demonstrated through examples of simplifying radicals like 3√24, 200t3u5, and 4√32w9x7.
Parallelogram is a quadrilateral with two pairs of parallel sides.
There are 6 properties of parallelogram.
1. A diagonal of a parallelogram divides it into two congruent triangles.
2. Opposites sides of a parallelogram are congruent.
3. Opposite angles of a parallelogram are congruent.
4. Consecutive angles of a parallelogram are supplementary.
5. If one angle in a parallelogram is right, then all angles are right.
6. The diagonals of a parallelogram bisect each other.
Parallelogram is a quadrilateral with two pairs of parallel sides.
There are 6 properties of parallelogram.
1. A diagonal of a parallelogram divides it into two congruent triangles.
2. Opposites sides of a parallelogram are congruent.
3. Opposite angles of a parallelogram are congruent.
4. Consecutive angles of a parallelogram are supplementary.
5. If one angle in a parallelogram is right, then all angles are right.
6. The diagonals of a parallelogram bisect each other.
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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
