Lesson 4: Operations on Radicals

New Concept

This lesson expands your skills by teaching you how to perform operations on radicals. You'll use product and quotient rules to multiply and divide them, and learn how to rationalize denominators to simplify fractional expressions.

What’s next

Next, you'll walk through worked examples for multiplying and dividing radicals, then apply these skills in a series of interactive practice cards and challenges.

Property

We have used the product rule for radicals to simplify square roots. We can also use the product rule to multiply radicals together.

Product Rule.
If $a, b \geq 0$, then $\sqrt{a}\sqrt{b} = \sqrt{ab}$

Examples

• To multiply $\sqrt{5} \cdot \sqrt{7}$, we use the product rule to get $\sqrt{5 \cdot 7} = \sqrt{35}$.

Property

We use the distributive law to remove parentheses from products involving radicals. Compare the product involving radicals on the left with the more familiar use of the distributive law on the right.

Examples

• To multiply $4(2y + 3\sqrt{5})$, distribute the 4: $4 \cdot 2y + 4 \cdot 3\sqrt{5} = 8y + 12\sqrt{5}$.

• To multiply $3\sqrt{2}(4 + \sqrt{5})$, distribute $3\sqrt{2}$: $3\sqrt{2} \cdot 4 + 3\sqrt{2} \cdot \sqrt{5} = 12\sqrt{2} + 3\sqrt{10}$.

Property

We use the quotient rule to simplify square roots of fractions by writing $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$. We can also use the quotient rule to simplify quotients of square roots.

Quotient Rule.
If $a \geq 0$ and $b > 0$, then $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$

Examples

• To simplify $\frac{\sqrt{48x^3}}{\sqrt{3x}}$, we use the quotient rule: $\sqrt{\frac{48x^3}{3x}} = \sqrt{16x^2} = 4x$.

Property

To simplify a fraction with a radical in the numerator, first simplify the square root. Then, factor the numerator and denominator to see if there are any common factors that can be divided out.

Caution: You cannot cancel a term in the numerator with the denominator, only a common factor. In $\frac{a+b}{a}$, you cannot cancel the $a$'s.

Examples

• To simplify $\frac{6 + \sqrt{18}}{3}$, first simplify the radical: $\frac{6 + 3\sqrt{2}}{3}$. Then factor the numerator: $\frac{3(2 + \sqrt{2})}{3} = 2 + \sqrt{2}$.

Property

The process of eliminating a radical from the denominator of a fraction is called rationalizing the denominator. This is done by multiplying the numerator and the denominator by the same root that appeared in the denominator originally. It is often best to simplify any radicals in the expression before attempting to rationalize.

Examples

• To rationalize $\frac{7}{\sqrt{3}}$, multiply the numerator and denominator by $\sqrt{3}$: $\frac{7 \cdot \sqrt{3}}{\sqrt{3} \cdot \sqrt{3}} = \frac{7\sqrt{3}}{3}$.

• To rationalize $\frac{6}{\sqrt{12}}$, first simplify the denominator: $\frac{6}{2\sqrt{3}} = \frac{3}{\sqrt{3}}$. Now rationalize: $\frac{3 \cdot \sqrt{3}}{\sqrt{3} \cdot \sqrt{3}} = \frac{3\sqrt{3}}{3} = \sqrt{3}$.