Section 6.2 Simplifying Radicals
Multiplying Radicals
√𝑎 ∙ √𝑏 = √𝑎𝑏
index must be the same
Example 1: If possible, multiply and then simplify.
a)
√2√5
b) √4 √16
c) 3√6 (4√6)
d) √2 √5
A radical expression is in simplified form if
1. There are no factors of the radicand that are
perfect squares under a square root, (4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 𝑥 , 𝑥 , ….…)
no perfect cubes under a cube root, (8, 27, 64, 125, 216, 𝑥 , 𝑥 , ….)
no perfect fourths under a fourth root, (16, 81, 256, 𝑥 , 𝑥 , …..)
no perfect fifths under a fifth root, and so on. (32, 243, 𝑥 , 𝑥 ……)
2. There are no fractions under a radical symbol.
3. There are no radicals in the denominator.
Multiplying Radicals Rule in reverse
√𝑎𝑏 = √𝑎 ∙ √𝑏
index must be the same
Example 2: Simplify the following radical expressions.
a)
√50
b) √50
c) √54
d) √54
e)
f) √40𝑎 𝑏
h)
32𝑥 𝑦
g)
48𝑥 𝑦
√−72𝑎 𝑏 𝑐 i)
243𝑥 𝑦 𝑧
Rationalizing a denominator: This means to get rid of a radical in a denominator. We do this by multiplying the fraction
by a radical of the same index to create a perfect square, cube, fourth, fifth, etc under the radical in the denominator, so
then the radical can be simplified away.
Dividing Radicals
=
√
√
index must be the same
Example 3: Write each of the following in simplified form that satisfies all three conditions.
a)
c)
√
b)