Radicals
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A simplified radical is one where there are no perfect squares inside the radical, no fractions
inside the radical, and no radicals in the denominator of a fraction.
Simplifying Radicals
1) Break the number or variable into smaller factors.
80 = 2 ∙ 2 ∙ 2 ∙ 2 ∙ 5
2) Find pairs (two of the same). One of them gets out, the other gets crossed out.
3) Whatever gets out of the radical, you multiply together. Whatever stays in the radical,
you multiply together.
2∙2 5
=4 5
Simplifying Radicals, Fractions Inside
1) See if you can reduce the fraction that is inside the radical so it is no longer is a
fraction. If yes, reduce then continue simplifying the radical.
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!
= 80 = 4 5
2) If no, break up the radical into two parts- the numerator and the denominator. See if
the denominator is a perfect square. If yes, then evaluate the denominator and continue
simplifying the numerator if necessary.
If no, see next.
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or
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5
Simplifying Radicals that are in the denominator
1) If there is a radical in the denominator of a fraction, can it be reduced with any radical
in the numerator? If so, reduce and continue.
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!
=5
!
= 5 3 done.
2) If there is a radical in the denominator of a fraction that won’t come out, then multiply
the numerator and denominator by the same radical in the denominator and simplify.
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∙
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=
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done.
Adding/Subtracting Radicals
1) Are they like radicals? (like radicals have the same number inside)
6 5 + 7 5 (Yes!)
2) If yes, add the outside numbers
6+7
5 = 13 5
3) If no, can you simplify each radical in order to make them like radicals?
8 + 32 = 2 ∙ 2 ∙ 2 + 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 = 2 2 + 4 2
4) Now they are like radicals. Follow step 2.
2 2 + 4 2 = 6 2 done.
Multiplying Radicals
1) To simplify, they do not need to be like radicals.
5 3∙4 2
2) Multiply the outside numbers, and multiply the inside numbers.
5∙4
3 ∙ 2 = 20 6
3) You may not be done. You may need to simplify the radical. In this case, you are
done.
Two Rules:
𝒂 ∙ 𝒃 = 𝒂𝒃
and
𝒂
𝒃
=
𝒂
𝒃