Simplifying Radical
Expressions
MGSE9-12.N.RN.2 Rewrite expressions involving
radicals. (i.e., simplify and/or use the operations
of addition, subtraction, and multiplication,
with radicals within expressions limited to
square roots).
Simplifying Radical Expressions
A radical expression is an expression that involves a square root.
EX:
√50
The radicand is the expression under the radical sign.
Simplest Radical Form
To simplify a radical expression:
1. Check the radicand for factors that are perfect squares.
2. Take the square root of the factors that are perfect squares.
9 is a perfect square
EX:
45
9𝑥5
Simplest Radical Form
3√5
Simplest Radical Form
You can also simplify a radical by looking at non-perfect square factors.
1. Factor the radicand as much as possible.
2. If you end up with two of the same factor, bring that factor out of the radical once.
48 can be factored to 2 x 24
24 can continue to break down to 2 x 12
12 can continue to break down to 2 x 6
6 can continue to break down to 2 x 3
Two 2s can be brought out of the radical
Simplest Radical Form
EX:
48
2 𝑥 24
2 𝑥 2 𝑥 12
2𝑥2𝑥2𝑥6
2𝑥2𝑥2𝑥2𝑥3
2 x 2√3
4√3
Note: There are other factors that you could start with, but they will all break down to the same thing.
You Try
3 24
5 √75
Adding and Subtracting Radicals
To add or subtract radical terms,
they must have the same
radicand. In other words, they
must be “like” terms.
Example:
5√5 - 7 5
(5 – 7) 5
To add or subtract radicals, just
add or subtract the coefficients in
front of the radical. DO NOT ADD
OR SUBTRACT THE RADICAND.
-2√5
Adding and Subtracting Radicals
If the terms do not have “like”
radicands, you must break the
radicals down as far as you can to
see if they will end up with like
radicands once they are
simplified.
Example:
2 20 - 45
2 4 ∗5- 9 ∗5
2*2 5-3 5
4 5-3 5
5
You Try
5 40 - 3 90
2 12 + 7 24
Multiplying Radicals
To multiply radicals, follow these
steps:
1. Multiply both the coefficients
and the radicands together even
if they are not “like” terms.
2. Simplify the radical if possible.
Example:
4 6 * 3 10
(4 * 3) 6 ∗ 10
12 60
12 6 ∗ 10
12 2 ∗ 3 ∗ 2 ∗ 5
12 * 2 3 ∗ 5
24 15
You Try
4 10 * 6 6
2 3 * 4 12