8.5 Divide Radical Expressions

Dividing Radical Expressions and Rationalizing Denominators

Dividing radical expressions and rationalizing denominators let you simplify fractions that contain roots. These techniques come up constantly when solving equations and simplifying expressions, so getting comfortable with them now pays off in later courses.

Division of Radical Expressions

The core rule for dividing radicals is straightforward: if two radicals share the same index, you can combine them under one radical sign by dividing the radicands.

$\frac{\sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[n]{\frac{a}{b}}$

For example: $\frac{\sqrt{50}}{\sqrt{2}} = \sqrt{\frac{50}{2}} = \sqrt{25} = 5$

When the indices match, follow these steps:

1. Divide the radicands (the numbers under the radical signs)
2. Keep the same index
3. Simplify the result if possible

After dividing, always check whether the radicand can be simplified. For instance, $\sqrt{8} = \sqrt{4 \cdot 2} = 2\sqrt{2}$.

When the indices are different, you need a common index before dividing:

1. Find the least common multiple (LCM) of the two indices
2. Rewrite each radical using the common index: $\sqrt[n]{a} = \sqrt[kn]{a^k}$, where $k$ is whatever factor converts the original index to the LCM
3. Now that the indices match, divide the radicands as usual
4. Simplify the result

Rationalizing Single Radical Denominators

A simplified expression should never have a radical in the denominator. Rationalizing means multiplying strategically to clear that radical out.

For a denominator with a single radical term, multiply the numerator and denominator by that same radical:

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

Here's a worked example:

1. Start with $\frac{6}{\sqrt{3}}$
2. Multiply top and bottom by $\sqrt{3}$: $\frac{6 \cdot \sqrt{3}}{\sqrt{3} \cdot \sqrt{3}} = \frac{6\sqrt{3}}{3}$
3. Simplify: $\frac{6\sqrt{3}}{3} = 2\sqrt{3}$

Don't forget to simplify the radicand too. If you end up with something like $\frac{2\sqrt{18}}{3}$, break it down: $\sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2}$, so the expression becomes $\frac{2 \cdot 3\sqrt{2}}{3} = 2\sqrt{2}$.

Rationalizing Two-Radical Denominators

When the denominator contains two terms with radicals (like $\sqrt{2} + \sqrt{3}$), multiplying by a single radical won't work. Instead, you multiply by the conjugate of the denominator.

The conjugate just flips the sign between the two terms: the conjugate of $a + b$ is $a - b$, and vice versa.

Why does this work? When you multiply conjugates, the cross terms cancel and you're left with a difference of squares, which eliminates the radicals:

$(\sqrt{5} + \sqrt{7})(\sqrt{5} - \sqrt{7}) = (\sqrt{5})^2 - (\sqrt{7})^2 = 5 - 7 = -2$

Steps to rationalize a two-radical denominator:

1. Identify the conjugate of the denominator
2. Multiply both the numerator and denominator by that conjugate
3. Apply the difference of squares pattern in the denominator
4. Simplify the numerator and reduce if possible

Worked example:

$\frac{1}{\sqrt{2} + \sqrt{3}} \cdot \frac{\sqrt{2} - \sqrt{3}}{\sqrt{2} - \sqrt{3}} = \frac{\sqrt{2} - \sqrt{3}}{(\sqrt{2})^2 - (\sqrt{3})^2} = \frac{\sqrt{2} - \sqrt{3}}{2 - 3} = \frac{\sqrt{2} - \sqrt{3}}{-1} = \sqrt{3} - \sqrt{2}$

Connecting Radicals to Fractional Exponents

Radicals can also be written using fractional exponents, and switching between the two forms is often useful for simplifying:

$\sqrt[n]{a} = a^{\frac{1}{n}}$

For example, $\sqrt[3]{x^2} = x^{\frac{2}{3}}$. This notation follows all the regular exponent rules, which can make division easier since $\frac{a^{m}}{a^{n}} = a^{m-n}$.