Name
class
date
Simplifying Radical Expressions
A radical is an expression of the form n x . The symbol used to show a radical is called the radical
sign. The expression under the radical sign is called the radicand. The number n is called the
index of the radical.
A radical is simplified if the radicand contains no factors that are powers equal to or greater than
the index. Use the product rule to simplify radicals. The product rule for radicals states that
n
abc 5 n a · n b · n c .
Example A
Simplify 32.
Step 1: Find the largest perfect square factor of 32, and
express 32 as a product.
32 5 16 ⋅ 2
Step 2: Write the result as the product of two radicals.
Then simplify. The power of 2 inside the
radical is 1, while the index of the radical is 2.
16 · 2 5 4 2
Solution:
32 5 4 2
Radicals can be added and subtracted only if they contain the same radicands. Check to see if the
radicals can be simplified before adding and subtracting because there may be common radicals
“hidden” inside the given expressions. Combining radicals is similar to combining like terms.
Combine the terms outside of the radical, but keep the common radicand.
Example B
Simplify 45 1 20 2 12 .
Step 1: Simplify each radical.
45 1 20 2 12 5 9 · 5 1 4 · 5 2 4 · 3 5
3 5 12 522 3
Step 2: Combine the radicals with same radicands.
3 5 12 5 22 3 55 5 22 3
Solution: 45 1 20 2 12 5 5 5 2 2 3
a
a
.
5
b
b
A fraction cannot contain a radical in its denominator. If a fraction contains a radical in its
denominator, the denominator must be rationalized by multiplying by an expression equal to 1.
The quotient rule for radicals states that
The expression used should lead to a radical in the denominator that is a perfect root. A fraction
d
with a denominator of d can be rationalized by multiplying the fraction by
.
d
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1
SpringBoard Precalculus, Unit 1
Simplifying Radical Expressions (continued)
Example C
3 .
5
Simplify
Step 1: Split the radical into a fraction of separate radicals.
3
3
5
5
5
Step 2: Multiply the fraction by 5 .
5
5
15
3
·
5
5
25
5
Step 3: Simplify.
15
15
5
5
25
Solution:
3
15
5
5
5
PRACTICE
Simplify each radical expression.
1.
200a 2 2.
3x · 8 x 3.
8 2 50 1 27 1 12 4.
6
5.
7.
(
3 25
)(
3 15
)
© 2015 College Board. All rights reserved.
2
12 2 2
6.
7 10
8.
4 xy y3
3 x3
(
)
SpringBoard Precalculus, Unit 1