The factoring square roots and cube roots page showed how to factor radicals that contain numbers.
But with simplifying radical expressions examples we can also factor radicals with variables as well as numbers.
Since radicals appear very often in Algebra topics, knowing how to factor with variables is also a key skill to learn.
Factoring a Radical Expression
A term like \sqrt{12} can be factored an simplified to 2\sqrt{3}.
As we’ll see the process is similar when variables are involved.
We can look at the radical expression \sqrt{9x^2}.
This radical can be factored and simplified, which actually eradicates the radical sign here.
\sqrt{9x^2} \space = \space \sqrt{9}\sqrt{x^2} \space = \space 3x
With factoring and simplifying rational expressions examples involving variables, the approach is the same as when working with just numbers.
We look for appropriate squares or cubes to factor out, and proceed with simplification.
Simplifying Radical Expressions Examples
(1.1)
Factor and simplify \sqrt{25x^6}.
Solution
We can factor out the simplest perfect square terms that make us the expression in the radicand.
\sqrt{25x^6} \space = \space \sqrt{25 \times x^2 \times x^2 \times x^2} \space = \space 5 \times \sqrt{x^2} \times \sqrt{x^2} \times \sqrt{x^2}
= \space 5 \times x \times x \times x \space = \space 5x^3
(1.2)
Factor and simplify \sqrt{8a^{3}b^{4}}.
Solution
In the same way as with (1.1), we factor out the simplest perfect square terms as much as we can.
\sqrt{8a^{3}b^{4}} \space = \space \sqrt{4 \times 2 \times a^2 \times a \times b^2 \times b^2}
= \space \sqrt{4} \times \sqrt{2} \times \sqrt{a^2} \times \sqrt{a} \times \sqrt{b^2} \times \sqrt{b^2}
= \space 2 \times \sqrt{2} \times a \times \sqrt{a} \times b \times b
= \space 2ab^2\sqrt{2a}
(1.3)
Factor and simplify \sqrt[3]{54x^{8}y^{3}z^{2}}.
Solution
Unlike the simplifying radical expressions examples in (1.1) and (1.2), this radical has an index of 3.
So we are looking to factor out the simplest perfect cubes as opposed to perfect squares.
\sqrt[3]{54x^{8}y^{3}z^{2}} \space = \space \sqrt[3]{27 \times 2} \times \sqrt[3]{x^3 \times x^3 \times x^2} \times \sqrt[3]{y^3} \times \sqrt[3]{z^2}
= \space \sqrt[3]{27} \times \sqrt[3]{2} \times \sqrt[3]{x^3} \times \sqrt[3]{x^3} \times \sqrt[3]{x^2} \times \sqrt[3]{y^3} \times \sqrt[3]{z^2}
= \space 3 \times \sqrt[3]{2} \times x \times x \times \sqrt[3]{x^2} \times y \times \sqrt[3]{z^2}
= \space 3x^{2}y \sqrt[3]{x^{2}z^{2}}