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.

\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*.

Looking at the pair [ 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}}

__Examples__

*(1.1)*Factor and simplify \sqrt{48}.

*Solution*Factor pairs of 48 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}}

*(1.2)*Factor and simplify \sqrt[3]{135}.

*Solution*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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