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# 11.4: Simplification of Radical Expressions

Difficulty Level: At Grade Created by: CK-12
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When you add and subtract radical expressions, combine radical terms only when they have the same expression under the radical sign. This is a lot like combining like terms in variable expressions.

#### Simplifying Expressions

1. Simplify the following expressions as much as possible.

a.) \begin{align*} 4 \sqrt{2} + 5 \sqrt{2}\end{align*}

\begin{align*}4 \sqrt{2} + 5 \sqrt{2} = 9 \sqrt{2}\end{align*}

b.) \begin{align*} 2 \sqrt{3} - \sqrt{2} + 5 \sqrt{3} + 10\sqrt{2}\end{align*}

\begin{align*}2 \sqrt{3} - \sqrt{2} + 5 \sqrt{3} + 10\sqrt{2} = 7 \sqrt{3} + 9 \sqrt{2}\end{align*}

It’s important to reduce all radicals to their simplest form in order to make sure that you’re combining all possible like terms in the expression. For example, the expression \begin{align*}\sqrt{8} - 2\sqrt{50}\end{align*} looks like it can’t be simplified any more because it has no like terms. However, when you write each radical in its simplest form, you get \begin{align*}2\sqrt{2} - 10 \sqrt{2}\end{align*}, and can combine those terms to get \begin{align*}-8 \sqrt{2}\end{align*}.

2. Simplify the following expressions as much as possible.

a) \begin{align*}4 \sqrt[3]{128} - \sqrt[3]{250}\end{align*}

\begin{align*}\text{Re-write radicals in simplest terms:} && & = 4 \sqrt[3]{2 \cdot 64} - \sqrt[3]{2 \cdot 125} = 16 \sqrt[3]{2} - 5 \sqrt[3]{2}\\ \text{Combine like terms:} && & = 11 \sqrt[3]{2}\end{align*}

b) \begin{align*}3 \sqrt{x^3} - 4x \sqrt{9x}\end{align*}

\begin{align*}\text{Re-write radicals in simplest terms:} && 3 \sqrt{x^2 \cdot x} - 12x \sqrt{x} & = 3x \sqrt{x} - 12x \sqrt{x}\\ \text{Combine like terms:} && & =-9x \sqrt{x}\end{align*}

When you multiply radical expressions, use the “raising a product to a power” rule: \begin{align*}\sqrt[m]{x \cdot y} = \sqrt[m]{x} \cdot \sqrt[m]{y}\end{align*}. In the case of the next example, apply this rule in reverse.

Simplify the expression \begin{align*}\sqrt6 \cdot \sqrt8.\end{align*}

\begin{align*}\sqrt{6} \cdot \sqrt{8} = \sqrt{6 \cdot 8} = \sqrt{48}\end{align*}

Or, in simplest radical form: \begin{align*}\sqrt{48} = \sqrt{16 \cdot 3} = 4 \sqrt{3}.\end{align*}

Another important fact is that \begin{align*}\sqrt{a} \cdot \sqrt{a} = \sqrt{a^2} = a.\end{align*}

When you multiply expressions that have numbers on both the outside and inside the radical sign, treat the numbers outside the radical sign and the numbers inside the radical sign separately. For example, \begin{align*}a \sqrt{b} \cdot c \sqrt{d} = ac \sqrt{bd}\end{align*}.

#### Multiplying Expressions

Multiply the following expressions.

Use distribution to eliminate the parentheses in each case.

a) \begin{align*}\sqrt{2}\left(\sqrt{3} + \sqrt{5}\right )\end{align*}

\begin{align*}\text{Distribute} \ \sqrt{2} \ \text{inside the parentheses:} && \sqrt{2}\left (\sqrt{3} + \sqrt{5}\right ) & = \sqrt{2} \cdot \sqrt{3} + \sqrt{2} \cdot \sqrt{5}\\ \text{Use the "raising a product to a power" rule:} && & = \sqrt{2 \cdot 3} + \sqrt{2 \cdot 5}\\ \text{Simplify:} && & =\sqrt{6} + \sqrt{10}\end{align*}

b) \begin{align*}2 \sqrt{x}\left (3 \sqrt{y} - \sqrt{x}\right )\end{align*}

\begin{align*}\text{Distribute} \ 2 \sqrt{x} \ \text{inside the parentheses:} && & =(2 \cdot 3)\left (\sqrt{x} \cdot \sqrt{y}\right ) - 2 \cdot \left ( \sqrt{x} \cdot \sqrt{x}\right )\\ \text{Multiply:} && & =6 \sqrt{xy} - 2 \sqrt{x^2}\\ \text{Simplify:} && & =6 \sqrt{xy} - 2x\end{align*}

c) \begin{align*}\left (2 + \sqrt{5}\right )\left (2 - \sqrt{6}\right )\end{align*}\begin{align*}\text{Distribute:} && (2 + \sqrt{5})(2 - \sqrt{6}) & = (2 \cdot 2) - \left (2 \cdot \sqrt{6}\right ) + \left( 2 \cdot \sqrt{5} \right ) - \left ( \sqrt{5} \cdot \sqrt{6} \right )\\ \text{Simplify:}&& & =4 - 2 \sqrt{6} + 2 \sqrt{5} - \sqrt{30}\end{align*}

d) \begin{align*}\left (2 \sqrt{x} + 1\right )\left (5 - \sqrt{x}\right )\end{align*}\begin{align*}\text{Distribute:} && \left (2 \sqrt{x} - 1\right )\left (5 - \sqrt{x}\right ) &=10 \sqrt{x} - 2x - 5 + \sqrt{x}\\ \text{Simplify:} && & =11 \sqrt{x} - 2x - 5\end{align*}

#### Rationalize the Denominator

Often, when you work with radicals, you end up with a radical expression in the denominator of a fraction. It’s traditional to write fractions in a form that doesn’t have radicals in the denominator, so you use a process called rationalizing the denominator to eliminate them.

Rationalizing is easiest when there’s just a radical and nothing else in the denominator, as in the fraction \begin{align*}\frac{2}{\sqrt{3}}\end{align*}. All you have to do then is multiply the numerator and denominator by a radical expression that makes the expression inside the radical into a perfect square, cube, or whatever power is appropriate. In the example above, multiply by \begin{align*}\sqrt{3}\end{align*}:

\begin{align*}\frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{2 \sqrt{3}}{3}\end{align*}

Cube roots and higher are a little trickier than square roots.

#### Rationalizing Expressions

Rationalize \begin{align*}\frac{7}{\sqrt[3]{5}}\end{align*}

You can’t just multiply by \begin{align*}\sqrt[3]{5}\end{align*}, because then the denominator would be \begin{align*}\sqrt[3]{5^2}\end{align*}. To make the denominator a whole number, you need to multiply the numerator and the denominator by \begin{align*}\sqrt[3]{5^2}\end{align*}:

\begin{align*}\frac{7}{\sqrt[3]{5}} \cdot \frac{\sqrt[3]{5^2}}{\sqrt[3]{5^2}} = \frac{7 \sqrt[3]{25}}{\sqrt[3]{5^3}} = \frac{7 \sqrt[3]{25}}{5}\end{align*}

Trickier still is when the expression in the denominator contains more than one term.

#### Rationalizing Multi-Term Denominators

Consider the expression \begin{align*}\frac{2}{2 + \sqrt{3}}\end{align*}. You can’t just multiply by \begin{align*}\sqrt{3}\end{align*}, because you’d have to distribute that term and then the denominator would be \begin{align*}2 \sqrt{3} + 3\end{align*}.

Instead, multiply by \begin{align*}2 - \sqrt{3}\end{align*}. This is a good choice because the product \begin{align*}\left (2 + \sqrt{3}\right )\left (2 - \sqrt{3}\right )\end{align*} is a product of a sum and a difference, which means it’s a difference of squares. The radicals cancel each other out when you distribute, and the denominator works out to a single number!\begin{align*}\left (2 + \sqrt{3} \right )\left (2 - \sqrt{3}\right ) = 2^2 - \left ( \sqrt{3}\right )^2 = 4 - 3 = 1\end{align*}

When you multiply both the numerator and denominator by \begin{align*}2 - \sqrt{3}\end{align*}, you end up with:

\begin{align*}\frac{2}{2 + \sqrt{3}} \cdot \frac{2 - \sqrt{3}}{2 - \sqrt{3}} = \frac{2\left (2 - \sqrt{3}\right )}{4 - 3} = \frac{4 - 2 \sqrt{3}}{1} = 4 - 2 \sqrt{3}\end{align*}

Now consider the expression \begin{align*}\frac{\sqrt{x} - 1}{\sqrt{x} - 2 \sqrt{y}}\end{align*}.

In order to eliminate the radical expressions in the denominator, multiply by \begin{align*}\sqrt{x} + 2 \sqrt{y}:\end{align*}

### Examples

Simplify the following expressions as much as possible.

#### Example 1

\begin{align*}4 \sqrt{3} + 2 \sqrt{12}\end{align*}

\begin{align*}\text{Simplify} \ \sqrt{12} && & =4 \sqrt{3} + 2 \sqrt{4 \cdot 3} = 4 \sqrt{3} + 4 \sqrt{3}\\ \text{Combine like terms:} && & =8 \sqrt{3}\end{align*}

#### Example 2

\begin{align*}10 \sqrt{24} - \sqrt{28}\end{align*}

\begin{align*}\text{Simplify} \ \sqrt{24} \ \text{and} \ \sqrt{28} && & =10 \sqrt{6 \cdot 4} - \sqrt{7 \cdot 4} = 20 \sqrt{6} - 2 \sqrt{7}\\ \text{There are no like terms.}\end{align*}

### Review

Simplify the following expressions as much as possible.

1. \begin{align*}3\sqrt{8} - 6 \sqrt{32}\end{align*}
2. \begin{align*}\sqrt{180} + \sqrt{405}\end{align*}
3. \begin{align*}\sqrt{6} - \sqrt{27} + 2 \sqrt{54} + 3 \sqrt{48}\end{align*}
4. \begin{align*}\sqrt{8x^3} - 4x \sqrt{98x}\end{align*}
5. \begin{align*}\sqrt{48a} + \sqrt{27a}\end{align*}
6. \begin{align*}\sqrt[3]{4x^3} + x \cdot \sqrt[3]{256}\end{align*}

Multiply the following expressions.

1. \begin{align*}\sqrt{6}\left (\sqrt{10} + \sqrt{8}\right )\end{align*}
2. \begin{align*}\left (\sqrt{a} - \sqrt{b}\right )\left (\sqrt{a} + \sqrt{b}\right )\end{align*}
3. \begin{align*}\left (2 \sqrt{x} + 5\right )\left (2 \sqrt{x} + 5\right )\end{align*}

Rationalize the denominator.

1. \begin{align*}\frac{7}{\sqrt{5}}\end{align*}
2. \begin{align*}\frac{9}{\sqrt{10}}\end{align*}
3. \begin{align*}\frac{2x}{\sqrt{5x}}\end{align*}
4. \begin{align*}\frac{\sqrt{5}}{\sqrt{3y}}\end{align*}
5. \begin{align*}\frac{12}{2 - \sqrt{5}}\end{align*}
6. \begin{align*}\frac{6 + \sqrt{3}}{4 - \sqrt{3}}\end{align*}
7. \begin{align*}\frac{\sqrt{x}}{\sqrt{2} + \sqrt{x}}\end{align*}
8. \begin{align*}\frac{5y}{2 \sqrt{y} - 5}\end{align*}

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Color Highlighted Text Notes

### Vocabulary Language: English

A radical expression is an expression with numbers, operations and radicals in it.

Rationalize the denominator

To rationalize the denominator means to rewrite the fraction so that the denominator no longer contains a radical.

Variable Expression

A variable expression is a mathematical phrase that contains at least one variable or unknown quantity.

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