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# Operations on Square Roots

### Learning Outcomes

• Add and subtract square roots.
• Rationalize denominators.
We can add or subtract radical expressions only when they have the same radicand and when they have the same radical type such as square roots. For example, the sum of $\sqrt{2}$ and $3\sqrt{2}$ is $4\sqrt{2}$. However, it is often possible to simplify radical expressions, and that may change the radicand. The radical expression $\sqrt{18}$ can be written with a $2$ in the radicand, as $3\sqrt{2}$, so $\sqrt{2}+\sqrt{18}=\sqrt{2}+3\sqrt{2}=4\sqrt{2}$.

### How To: Given a radical expression requiring addition or subtraction of square roots, solve.

1. Simplify each radical expression.
2. Add or subtract expressions with equal radicands.

### Example: Adding Square Roots

Add $5\sqrt{12}+2\sqrt{3}$.

Answer: We can rewrite $5\sqrt{12}$ as $5\sqrt{4\cdot 3}$. According the product rule, this becomes $5\sqrt{4}\sqrt{3}$. The square root of $\sqrt{4}$ is 2, so the expression becomes $5\left(2\right)\sqrt{3}$, which is $10\sqrt{3}$. Now we can the terms have the same radicand so we can add.

$10\sqrt{3}+2\sqrt{3}=12\sqrt{3}$

### Try It

Add $\sqrt{5}+6\sqrt{20}$.

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Watch this video to see more examples of adding roots. https://youtu.be/S3fGUeALy7E

### Example: Subtracting Square Roots

Subtract $20\sqrt{72{a}^{3}{b}^{4}c}-14\sqrt{8{a}^{3}{b}^{4}c}$.

Answer: Rewrite each term so they have equal radicands.

\begin{align} 20\sqrt{72{a}^{3}{b}^{4}c}& = 20\sqrt{9}\sqrt{4}\sqrt{2}\sqrt{a}\sqrt{{a}^{2}}\sqrt{{\left({b}^{2}\right)}^{2}}\sqrt{c} \\ & = 20\left(3\right)\left(2\right)a{b}^{2}\sqrt{2ac} \\ & = 120a{b}^{2}\sqrt{2ac}\\ \text{ } \end{align}
\begin{align} 14\sqrt{8{a}^{3}{b}^{4}c}& = 14\sqrt{2}\sqrt{4}\sqrt{a}\sqrt{{a}^{2}}\sqrt{{\left({b}^{2}\right)}^{2}}\sqrt{c} \\ & = 14\left(2\right)a{b}^{2}\sqrt{2ac} \\ & = 28a{b}^{2}\sqrt{2ac} \end{align}
Now the terms have the same radicand so we can subtract.
$120a{b}^{2}\sqrt{2ac}-28a{b}^{2}\sqrt{2ac}=92a{b}^{2}\sqrt{2ac} \\$
Note that we do not need an absolute value around the a because the $a^3$ under the radical means that a can't be negative.

### Try It

Subtract $3\sqrt{80x}-4\sqrt{45x}$.

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in the next video we show more examples of how to subtract radicals. https://youtu.be/77TR9HsPZ6M

## Rationalize Denominators

### Recall the identity property of addition

We leverage an important and useful identity in this section in a technique commonly used in college algebra:

rewriting an expression by multiplying it by a well-chosen form of the number 1.

Because the additive identity states that $a\cdot1=a$, we are able to multiply the top and bottom of any fraction by the same number without changing its value. We use this idea when we rationalize the denominator.
When an expression involving square root radicals is written in simplest form, it will not contain a radical in the denominator. We can remove radicals from the denominators of fractions using a process called rationalizing the denominator. We know that multiplying by 1 does not change the value of an expression. We use this property of multiplication to change expressions that contain radicals in the denominator. To remove radicals from the denominators of fractions, multiply by the form of 1 that will eliminate the radical. For a denominator containing a single term, multiply by the radical in the denominator over itself. In other words, if the denominator is $b\sqrt{c}$, multiply by $\dfrac{\sqrt{c}}{\sqrt{c}}$. For a denominator containing the sum or difference of a rational and an irrational term, multiply the numerator and denominator by the conjugate of the denominator, which is found by changing the sign of the radical portion of the denominator. If the denominator is $a+b\sqrt{c}$, then the conjugate is $a-b\sqrt{c}$.

### How To: Given an expression with a single square root radical term in the denominator, rationalize the denominator.

1. Multiply the numerator and denominator by the radical in the denominator.
2. Simplify.

### Example: Rationalizing a Denominator Containing a Single Term

Write $\dfrac{2\sqrt{3}}{3\sqrt{10}}$ in simplest form.

Answer: The radical in the denominator is $\sqrt{10}$. So multiply the fraction by $\dfrac{\sqrt{10}}{\sqrt{10}}$. Then simplify.

\begin{align}\frac{2\sqrt{3}}{3\sqrt{10}}\cdot \frac{\sqrt{10}}{\sqrt{10}} &= \frac{2\sqrt{30}}{30} \\ &= \frac{\sqrt{30}}{15}\end{align}

### Try It

Write $\dfrac{12\sqrt{3}}{\sqrt{2}}$ in simplest form.

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### How To: Given an expression with a radical term and a constant in the denominator, rationalize the denominator.

1. Find the conjugate of the denominator.
2. Multiply the numerator and denominator by the conjugate.
3. Use the distributive property.
4. Simplify.

### Example: Rationalizing a Denominator Containing Two Terms

Write $\dfrac{4}{1+\sqrt{5}}$ in simplest form.

Answer: Begin by finding the conjugate of the denominator by writing the denominator and changing the sign. So the conjugate of $1+\sqrt{5}$ is $1-\sqrt{5}$. Then multiply the fraction by $\dfrac{1-\sqrt{5}}{1-\sqrt{5}}$.

\begin{align}\frac{4}{1+\sqrt{5}}\cdot \frac{1-\sqrt{5}}{1-\sqrt{5}} &= \frac{4 - 4\sqrt{5}}{-4} && \text{Use the distributive property}. \\ &=\sqrt{5}-1 && \text{Simplify}. \end{align}

### Try It

Write $\dfrac{7}{2+\sqrt{3}}$ in simplest form.

Answer: [latex-display]14 - 7\sqrt{3}[/latex-display]

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