9.5 Divide Square Roots

Division and Simplification of Square Roots

Dividing square roots follows a clean rule: when two square roots are divided, you can combine them under a single radical. This section covers that rule, plus how to rationalize denominators so no square roots remain on the bottom of a fraction.

Division with Square Roots

The core rule for dividing square roots is the Quotient Rule for Radicals:

$\sqrt{a} \div \sqrt{b} = \sqrt{\frac{a}{b}}$

As long as both square roots have the same index, you can divide the radicands (the numbers under the radical) directly.

Example: $\sqrt{18} \div \sqrt{2} = \sqrt{\frac{18}{2}} = \sqrt{9} = 3$

A few other situations you'll see:

• Dividing a number by a square root: Rewrite the number as a square root first, then apply the rule. $4 \div \sqrt{2} = \sqrt{16} \div \sqrt{2} = \sqrt{\frac{16}{2}} = \sqrt{8} = 2\sqrt{2}$

• A square root divided by itself always equals 1: $\sqrt{a} \div \sqrt{a} = 1$

• After dividing, always check whether the result can be simplified further by looking for perfect square factors.

Rationalizing Single-Term Denominators

A rationalized denominator contains no square roots. To rationalize a fraction with a single square root in the denominator, multiply both the numerator and denominator by that square root.

Steps:

1. Identify the square root in the denominator.
2. Multiply the numerator and denominator by that same square root.
3. Simplify the denominator (since $\sqrt{b} \cdot \sqrt{b} = b$).
4. Reduce the fraction if possible.

Example 1:

$\frac{2}{\sqrt{3}} = \frac{2 \cdot \sqrt{3}}{\sqrt{3} \cdot \sqrt{3}} = \frac{2\sqrt{3}}{3}$

Example 2 (with simplifying):

$\frac{3}{\sqrt{12}} = \frac{3 \cdot \sqrt{12}}{\sqrt{12} \cdot \sqrt{12}} = \frac{3\sqrt{12}}{12}$

Now simplify $\sqrt{12} = 2\sqrt{3}$, so:

$\frac{3 \cdot 2\sqrt{3}}{12} = \frac{6\sqrt{3}}{12} = \frac{\sqrt{3}}{2}$

Techniques for Two-Term Denominators

When the denominator has two terms (like $2 + \sqrt{3}$), multiplying by just the square root won't clear it. Instead, you multiply by the conjugate of the denominator.

The conjugate flips the sign between the two terms:

• The conjugate of $a + \sqrt{b}$ is $a - \sqrt{b}$
• The conjugate of $a - \sqrt{b}$ is $a + \sqrt{b}$

This works because of the difference of squares pattern: $(a + \sqrt{b})(a - \sqrt{b}) = a^2 - b$, which eliminates the square root from the denominator.

Steps:

1. Identify the conjugate of the denominator.
2. Multiply both numerator and denominator by the conjugate.
3. Apply the difference of squares in the denominator.
4. Distribute in the numerator and simplify.

Example 1:

$\frac{1}{2 + \sqrt{3}} \cdot \frac{2 - \sqrt{3}}{2 - \sqrt{3}} = \frac{2 - \sqrt{3}}{4 - 3} = 2 - \sqrt{3}$

Example 2:

$\frac{2}{1 + \sqrt{5}} \cdot \frac{1 - \sqrt{5}}{1 - \sqrt{5}} = \frac{2(1 - \sqrt{5})}{1 - 5} = \frac{2 - 2\sqrt{5}}{-4} = \frac{-1 + \sqrt{5}}{2}$

Additional Concepts in Square Root Division

• Perfect square factors help you simplify after dividing. For instance, $\sqrt{16x^2} = 4x$ (when $x$ is positive) because both 16 and $x^2$ are perfect squares.
• Always simplify your final answer fully: reduce fractions and simplify any remaining radicals.