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}}