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Factoring Radicals, Factoring Cube Roots
and Square Roots

Knowing how to factor radicals properly is important when it comes to dealing with factoring cube roots and square roots involving numbers.

A particularly helpful fact to remember is that,   \sqrt{a} \times \sqrt{a} \space = \space \sqrt{a \times a }.

Which is that the square root of two numbers multiplied together, is the same as there individual square roots multiplied together.

This also applies to situations where we are factoring cube roots.

\sqrt[3]{a} \times \sqrt[3]{a} \space = \space \sqrt[3]{a \times a }

Factoring a Square Root:

We can consider the square root radical term   \sqrt{12}.

If we first think about  12  on its own as a number.
We can list the factor pairs, the numbers that multiplied together make   12.

12,1      6,2      4,3

So we have two real possibilities to rewrite the square root.

\sqrt{6 \times 2} \space \space , \space \space \sqrt{4 \times 3}

“Perfect squares” are numbers that have a square root which is a whole number,
so we initially look for a factor pair that contains a perfect square.

With  [ 4,3 ]4  is a perfect square.

So:   \sqrt{12} \space = \space \sqrt{4 \times 3} \space = \space \sqrt{4} \times \sqrt{3} \space = \space 2 \times \sqrt{3} \space = \space 2{\sqrt{3}}

With factoring and simplifying radical expressions examples, we also want to get the simplest number or term we can as the radicand inside the radical.
So a factor square with the largest perfect square possible is what we primarily look for when factoring square roots.

Factoring Cube Roots:

The principle is the same with factoring cube roots, but slightly different.

Instead of looking for factors that are perfect squares, we look for “perfect cubes”.

For example with   \sqrt[3]{40}.

This cube root can factor to   \sqrt[3]{8 \times 5}.

Which is handy as  8  is a perfect cube.

\sqrt[3]{8 \times 5} \space = \space \sqrt[3]{8} \times \sqrt[3]{5} \space = \space 2 \times \sqrt[3]{5} \space = \space 2{\sqrt[3]{5}}



Factor and simplify  \sqrt{48}.


Factor pairs of &nbsp48  are:   48,1      24,2      12,4      8,6      16,3.

The number  16  is a perfect square, and the largest one among the factor pairs.

\sqrt{48} \space = \space \sqrt{16 \times 3} \space = \space \sqrt{16} \times \sqrt{3} \space = \space 4 \times \sqrt{3} \space = \space 4{\sqrt{3}}


Factor and simplify   \sqrt[3]{135}.


This example requires factoring and simplifying a cube root.
So we need to look for a factor pair containing a perfect cube, instead of a perfect square.

Factor pairs of  135  are:   135,1      45,3      27,5      15,9.

The number  27  is a perfect cube, and is the only one among the factor pairs.

\sqrt{135} \space = \space \sqrt[3]{27 \times 5} \space = \space \sqrt[3]{27} \times \sqrt[3]{5} \space = \space 3 \times \sqrt[3]{5} \space = \space 3{\sqrt[3]{5}}

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