Radicals
The symbol √ is a radical sign. The number or expression inside the radical sign
is the radicand. √𝑎 is a radical.
A radical can be simplified whenever the radicand has a factor that is a perfect
square.
To simplify:
1. Use a factor tree to factor the radicand.
2. Look for pairs of identical factors.
3. Remove the factor of identical pairs only
Example:
√48 = √(2 ∙ 2) ∙ (2 ∙ 2) ∙ 3 = 2 ∙ 2√3 = 4√3
Properties of Radicals
(𝑎 > 0, 𝑏 > 0)
1. Product Property: √𝑎𝑏 = √𝑎 ∙ √𝑏
𝑎
√𝑎
𝑏
√𝑏
2. Quotient Property: √ =
A radical is in simplest form when:
1. The radicand contains no factor that is a perfect square.
2. The radicand does not contain a fraction.
3. No radical appears in the denominator.
Rationalizing the denominator is the process of eliminating square root in the
denominator of a fraction.
To rationalize the denominator:
1. Simplify the radicals in both the numerator and the denominator of the
fraction.
2. Multiply both the numerator and denominator by the simplified radical in the
denominator.
Examples:
1.
√7
√3
2.
18
√3
=
=
√7 √3
∙
√3 √3
18
√3
√3 √3
∙
=
=
√21
√9
=
18√3
√9
√21
3
=
18√3
3
= 6 √3
Addition and Subtraction of Radicals
Only like radicals can be combined.
Like Radicals
√5, 6√5
Unlike Radicals
√5, √3
Examples:
1. 3√5 + 6√2 − √5 = 2√5 + 6√2
2. √12 + 4√3 = √(2 ∙ 2) ∙ 3 + 4√3 = 2√3 + 4√3 = 6√3