Lesson 9.5: Divide Square Roots

New Concept

This lesson covers dividing square roots by using the Quotient Property. You will also learn to rationalize denominators, a crucial technique for removing radicals from the denominator of a fraction, ensuring the expression is fully simplified.

What’s next

Next, you’ll tackle interactive examples for dividing square roots. Then, you'll move on to practice cards focused on rationalizing both one-term and two-term denominators.

Property

We simplify fractions by removing factors common to the numerator and the denominator.
When we have a fraction with a square root in the numerator, we first simplify the square root.
Then we can look for common factors.

Examples

• To simplify $\frac{\sqrt{48}}{8}$, first simplify the radical: $\frac{\sqrt{16 \cdot 3}}{8} = \frac{4\sqrt{3}}{8}$. Then, cancel the common factor of 4 to get $\frac{\sqrt{3}}{2}$.

• For $\frac{8-\sqrt{32}}{12}$, simplify the radical: $\frac{8-\sqrt{16 \cdot 2}}{12} = \frac{8-4\sqrt{2}}{12}$. Factor out 4 in the numerator $\frac{4(2-\sqrt{2})}{12}$ and cancel to get $\frac{2-\sqrt{2}}{3}$.

Property

If $a, b$ are non-negative real numbers and $b \neq 0$, then

We use the Quotient Property of Square Roots 'in reverse' when the fraction we start with is the quotient of two square roots, and neither radicand is a perfect square. This may reveal common factors.

Examples

• To simplify $\frac{\sqrt{50}}{\sqrt{98}}$, we combine them: $\sqrt{\frac{50}{98}} = \sqrt{\frac{25 \cdot 2}{49 \cdot 2}} = \sqrt{\frac{25}{49}} = \frac{5}{7}$.

• For $\frac{\sqrt{48y^5}}{\sqrt{3y}}$, combine to get $\sqrt{\frac{48y^5}{3y}} = \sqrt{16y^4}$. Then simplify the radical to get $4y^2$.

Property

Rationalizing the Denominator
The process of converting a fraction with a radical in the denominator to an equivalent fraction whose denominator is an integer is called rationalizing the denominator.

Simplified Square Roots
A square root is considered simplified if there are

• no perfect square factors in the radicand
• no fractions in the radicand
• no square roots in the denominator of a fraction

Examples

• To rationalize $\frac{5}{\sqrt{2}}$, multiply the top and bottom by $\sqrt{2}$: $\frac{5 \cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}} = \frac{5\sqrt{2}}{2}$.

Property

When the denominator of a fraction is a sum or difference with square roots, we use the Product of Conjugates pattern to rationalize the denominator.

When we multiply a binomial that includes a square root by its conjugate, the product has no square roots.

Examples

• To simplify $\frac{3}{5+\sqrt{7}}$, multiply by the conjugate $(5-\sqrt{7})$: $\frac{3(5-\sqrt{7})}{(5+\sqrt{7})(5-\sqrt{7})} = \frac{3(5-\sqrt{7})}{25-7} = \frac{3(5-\sqrt{7})}{18} = \frac{5-\sqrt{7}}{6}$.

• Simplify $\frac{6}{4-\sqrt{10}}$ by multiplying by $(4+\sqrt{10})$: $\frac{6(4+\sqrt{10})}{(4-\sqrt{10})(4+\sqrt{10})} = \frac{6(4+\sqrt{10})}{16-10} = \frac{6(4+\sqrt{10})}{6} = 4+\sqrt{10}$.