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# Simplify radical expressions

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

$$\beginpmatrix xy \endpmatrix^a=x^ay^a$$

$$\beginpmatrix \fracxy \endpmatrix^a=\fracx^ay^a $$

We also know that

$$\sqrt[a]x=x^\frac1a$$

$$or$$

$$\sqrtx=x^\frac12$$

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.

$$\sqrtxy=\sqrtx\cdot \sqrty$$

$$\sqrt\fracxy=\frac\sqrtx\sqrty$$

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

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

$$\sqrtx^2=x\: \: where\: \: x\geq 0$$

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

$$\sqrt16x=\sqrt16\cdot \sqrtx=\sqrt4^2\cdot \sqrtx=4\sqrtx$$

no fractions in the radicand and

$$\sqrt\frac2516x^2=\frac\sqrt25\sqrt16\cdot \sqrtx^2=\frac54x$$

no radicals appear in the denominator of a fraction.

$$\sqrt\frac1516=\frac\sqrt15\sqrt16=\frac\sqrt154$$

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.

$$\sqrt\fracxy=\frac\sqrtx\sqrty\cdot \colorgreen \frac\sqrty\sqrty=\frac\sqrtxy\sqrty^2=\frac\sqrtxyy$$

Binomials like

$$x\sqrty+z\sqrtw\: \: and\: \: x\sqrty-z\sqrtw$$

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.

$$\fracx4+\sqrtx=\fracx\left ( \colorgreen 4-\sqrtx \right )\left ( 4+\sqrtx \right )\left ( \colorgreen 4-\sqrtx \right )= $$

$$=\fracx\left ( 4-\sqrtx \right )16-\left ( \sqrtx \right )^2=\frac4x-x\sqrtx16-x$$

**Video lesson**

Simplify the radical expression

$$\fracx5-\sqrtx$$

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