Simplify radical expressions

The properties of exponents, which we've talked about earlier, tell us among other things that

$$\begin{pmatrix} xy \end{pmatrix}^{a}=x^{a}y^{a}$$

$$\begin{pmatrix} \frac{x}{y} \end{pmatrix}^{a}=\frac{x^{a}}{y^{a}} $$

We also know that




If we combine these two things then we get the product property of radicals and the quotient property of radicals. These two properties tell us that the square root of a product equals the product of the square roots of the factors.

$$\sqrt{xy}=\sqrt{x}\cdot \sqrt{y}$$


$$where\:\: x\geq 0,y\geq 0 $$

The answer can't be negative and x and y can't be negative since we then wouldn't get a real answer. In the same way we know that

$$\sqrt{x^{2}}=x\: \: where\: \: x\geq 0$$

These properties can be used to simplify radical expressions. A radical expression is said to be in its simplest form if there are

no perfect square factors other than 1 in the radicand

$$\sqrt{16x}=\sqrt{16}\cdot \sqrt{x}=\sqrt{4^{2}}\cdot \sqrt{x}=4\sqrt{x}$$

no fractions in the radicand and

$$\sqrt{\frac{25}{16}x^{2}}=\frac{\sqrt{25}}{\sqrt{16}}\cdot \sqrt{x^{2}}=\frac{5}{4}x$$

no radicals appear in the denominator of a fraction.


If the denominator is not a perfect square you can rationalize the denominator by multiplying the expression by an appropriate form of 1 e.g.

$$\sqrt{\frac{x}{y}}=\frac{\sqrt{x}}{\sqrt{y}}\cdot {\color{green} {\frac{\sqrt{y}}{\sqrt{y}}}}=\frac{\sqrt{xy}}{\sqrt{y^{2}}}=\frac{\sqrt{xy}}{y}$$

Binomials like

$$x\sqrt{y}+z\sqrt{w}\: \: and\: \: x\sqrt{y}-z\sqrt{w}$$

are called conjugates to each other. The product of two conjugates is always a rational number which means that you can use conjugates to rationalize the denominator e.g.

$$\frac{x}{4+\sqrt{x}}=\frac{x\left ( {\color{green} {4-\sqrt{x}}} \right )}{\left ( 4+\sqrt{x} \right )\left ( {\color{green}{ 4-\sqrt{x}}} \right )}= $$

$$=\frac{x\left ( 4-\sqrt{x} \right )}{16-\left ( \sqrt{x} \right )^{2}}=\frac{4x-x\sqrt{x}}{16-x}$$

Video lesson

Simplify the radical expression