Algebra 1 · Unit 6
Clear a square root or cube root from a monomial denominator.
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A monomial leftover in the denominator
If the denominator is √a or k√a, multiply top and bottom by √a.
The product of conjugates is not needed here
This page is a single radical in the denominator, not a binomial like 1 + √2.
Cube roots
For ∛a in the denominator, multiply by enough cube roots so the insides make a perfect cube.
Simplify first
Pull perfect squares or cubes out of the radical before you rationalize, when you can.
One square root in the denominator.
Multiply top and bottom by √3.
√33
Answer: √33
√22
Answer: √22
5√55 = √5
Answer: √5
2√77
Answer: 2√77
3√66 = √62
Answer: √62
√32 × 3 = √36
Answer: √36
Simplify the radical first, or a small coefficient.
√8 = 2√2
12√2
√24
Answer: √24
√12 = 2√3
62√3 = 3√3
3√33 = √3
Answer: √3
2√5 = 2√55
Answer: 2√55
x√33
Answer: x√33
Multiply by ∛4 so the insides are 8.
∛42
Answer: ∛42
√20 = 2√5
52√5 = 5√510 = √52
Answer: √52
A leftover after simplifying, or a cube root.
√18 = 3√2
33√2 = 1√2
√22
Answer: √22
2x2√2 = x√2
x√22
Answer: x√22
4 × 2 = 8, so multiply by ∛2.
∛22
Answer: ∛22
√12 = 2√3
63 × 2√3 = 1√3
√33
Answer: √33
√3√15 = √315 = √15
√55
Answer: √55
Multiply by ∛9.
∛92 × 3 = ∛96
Answer: ∛96
√2√2 = 1
Multiplying by 1 leaves the number the same. The new form √22 has a rational denominator.
Answer: multiply by 1
√88 = 2√28 = √24
Faster: √8 = 2√2 first, then √24.
Answer: √24
\(a^{\frac{1}{3}}\) × \(a^{\frac{2}{3}}\) = \(a^{\frac{3}{3}}\) = a
Multiply top and bottom by ∛a².
Answer: a² under the cube root
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