# Factoring and SimplifyingRadical Expressions Examples

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.

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}.

\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.

(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}}

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