### Learning Objectives

In this section students will:

A hardware store sells 16-ft ladders and 24-ft ladders. A window is located 12 feet above the ground. A ladder needs to be purchased that will reach the window from a point on the ground 5 feet from the building. To find out the length of ladder needed, we can draw a right triangle as shown in Figure 1, and use the Pythagorean Theorem.

$\begin{array}{ccc}\hfill {a}^{2}+{b}^{2}& =& {c}^{2}\hfill \\ \hfill {5}^{2}+{12}^{2}& =& {c}^{2}\hfill \\ \hfill 169& =& {c}^{2}\hfill \end{array}$

Now, we need to find out the length that, when squared, is 169, to determine which ladder to choose. In other words, we need to find a square root. In this section, we will investigate methods of finding solutions to problems such as this one.

### 3.2.1 – Evaluating Square Roots

When the square root of a number is squared, the result is the original number. Since $\,{4}^{2}=16,$ the square root of $\,16\,$ is $\,4.\,$ The square root function is the inverse of the squaring function just as subtraction is the inverse of addition. To undo squaring, we take the square root.

In general terms, if $\,a\,$ is a positive real number, then the square root of $\,a\,$ is a number that, when multiplied by itself, gives $\,a.\,$ The square root could be positive or negative because multiplying two negative numbers gives a positive number. The principal square root is the nonnegative number that when multiplied by itself equals $\,a.\,$ The square root obtained using a calculator is the principal square root.

The principal square root of $\,a\,$ is written as $\,\sqrt{a}.\,$ The symbol is called a radical, the term under the symbol is called the radicand, and the entire expression is called a radical expression.

### Principal Square Root

The principal square root of $\,a\,$ is the nonnegative number that, when multiplied by itself, equals $\,a.\,$ It is written as a radical expression, with a symbol called a radical over the term called the radicand: $\,\sqrt{a}.$

Does $\,\sqrt{25}=±5?$

No. Although both $\,{5}^{2}\,$ and $\,{\left(-5\right)}^{2}\,$ are $\,25,$ the radical symbol implies only a nonnegative root, the principal square root. The principal square root of 25 is $\,\sqrt{25}=5.$

### Example 1 – Evaluating Square Roots

Evaluate each expression.

1. $\sqrt{100}$
2. $\sqrt{\sqrt{16}}$
3. $\sqrt{25+144}$
4. $\sqrt{49}-\sqrt{81}$
1. $\sqrt{100}=10\,$ because $\,{10}^{2}=100$
2. $\sqrt{\sqrt{16}}=\sqrt{4}=2\,$ because $\,{4}^{2}=16\,$ and $\,{2}^{2}=4$
3. $\sqrt{25+144}=\sqrt{169}=13\,$ because $\,{13}^{2}=169$
4. $\sqrt{49}-\sqrt{81}=7-9=-2\,$ because $\,{7}^{2}=49\,$ and $\,{9}^{2}=81$

For $\,\sqrt{25+144},$ can we find the square roots before adding?

No. $\,\sqrt{25}+\sqrt{144}=5+12=17.\,$ This is not equivalent to $\,\sqrt{25+144}=13.\,$ The order of operations requires us to add the terms in the radicand before finding the square root.

### Try It

Evaluate each expression.

1. $\sqrt{225}$
2. $\sqrt{\sqrt{81}}$
3. $\sqrt{25-9}$
4. $\sqrt{36}+\sqrt{121}$
1. $15$
2. $3$
3. $4$
4. $17$

### 3.2.2 – Using the Product Rule to Simplify Square Roots

To simplify a square root, we rewrite it such that there are no perfect squares in the radicand. There are several properties of square roots that allow us to simplify complicated radical expressions. The first rule we will look at is the product rule for simplifying square roots, which allows us to separate the square root of a product of two numbers into the product of two separate rational expressions. For instance, we can rewrite $\,\sqrt{15}\,$ as $\,\sqrt{3}\cdot \sqrt{5}.\,$ We can also use the product rule to express the product of multiple radical expressions as a single radical expression.

### The Product Rule for Simplifying Square Roots

If $\,a\,$ and $\,b\,$ are nonnegative, the square root of the product $\,ab\,$ is equal to the product of the square roots of $\,a\,$ and $\,b.\,$

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

### How To

Given a square root radical expression, use the product rule to simplify it.

1. Factor any perfect squares from the radicand.
3. Simplify.

### Example 2 – Using the Product Rule to Simplify Square Roots

1. $\sqrt{300}$
2. $\sqrt{162{a}^{5}{b}^{4}}$
1. $\begin{array}{cc}\sqrt{100\cdot 3}\hfill & \phantom{\rule{4.5em}{0ex}}\text{Factor perfect square from radicand}.\hfill \\ \sqrt{100}\cdot \sqrt{3}\hfill & \phantom{\rule{4.5em}{0ex}}\text{Write radical expression as product of radical expressions}.\hfill \\ 10\sqrt{3}\hfill & \phantom{\rule{4.5em}{0ex}}\text{Simplify}.\hfill \end{array}$
2. $\begin{array}{cc}\sqrt{81{a}^{4}{b}^{4}\cdot 2a}\hfill & \phantom{\rule{2em}{0ex}}\text{Factor perfect square from radicand}.\hfill \\ \sqrt{81{a}^{4}{b}^{4}}\cdot \sqrt{2a}\hfill & \phantom{\rule{2em}{0ex}}\text{Write radical expression as product of radical expressions}.\hfill \\ 9{a}^{2}{b}^{2}\sqrt{2a}\hfill & \phantom{\rule{2em}{0ex}}\text{Simplify}.\hfill \end{array}$

### Try It

Simplify $\,\sqrt{50{x}^{2}{y}^{3}z}.$

$5|x||y|\sqrt{2yz}.\,$ Notice the absolute value signs around x and y? That’s because their value must be positive!

### How To

Given the product of multiple radical expressions, use the product rule to combine them into one radical expression.

1. Express the product of multiple radical expressions as a single radical expression.
2. Simplify.

### Example 3 – Using the Product Rule to Simplify the Product of Multiple Square Roots

Simplify the radical expression $\sqrt{12}\cdot\sqrt{3}$.

$\begin{array}{cc}\sqrt{12\cdot 3}\hfill & \phantom{\rule{2em}{0ex}}\text{Express the product as a single radical expression}.\hfill \\ \sqrt{36}\hfill & \phantom{\rule{2em}{0ex}}\text{Simplify}.\hfill \\ 6\hfill & \end{array}$

### Try It

Simplify $\,\sqrt{50x}\cdot \sqrt{2x}\,$ assuming $\,x>0.$

$10|x|$

### 3.2.3 – Using the Quotient Rule to Simplify Square Roots

Just as we can rewrite the square root of a product as a product of square roots, so too can we rewrite the square root of a quotient as a quotient of square roots, using the quotient rule for simplifying square roots. It can be helpful to separate the numerator and denominator of a fraction under a radical so that we can take their square roots separately. We can rewrite $\,\sqrt{\frac{5}{2}}\,$ as $\,\frac{\sqrt{5}}{\sqrt{2}}.$

### The Quotient Rule for Simplifying Square Roots

The square root of the quotient $\,\frac{a}{b}\,$ is equal to the quotient of the square roots of $\,a\,$ and $\,b,$ where $\,b\ne 0.$

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

### How To

Given a radical expression, use the quotient rule to simplify it.

1. Write the radical expression as the quotient of two radical expressions.
2. Simplify the numerator and denominator.

### Example 4 – Using the Quotient Rule to Simplify Square Roots

$\sqrt{\frac{5}{36}}$

$\begin{array}{cc}\frac{\sqrt{5}}{\sqrt{36}}\hfill & \phantom{\rule{2em}{0ex}}\text{Write as quotient of two radical expressions}.\hfill \\ \frac{\sqrt{5}}{6}\hfill & \phantom{\rule{2em}{0ex}}\text{Simplify denominator}.\hfill \end{array}$

### Try It

Simplify $\,\sqrt{\frac{2{x}^{2}}{9{y}^{4}}}.$

$\frac{x\sqrt{2}}{3{y}^{2}}.\,$ We do not need the absolute value signs for $\,{y}^{2}\,$ because that term will always be nonnegative.

### Example 5 – Using the Quotient Rule to Simplify an Expression with Two Square Roots

$\frac{\sqrt{234{x}^{11}y}}{\sqrt{26{x}^{7}y}}$

$\begin{array}{cc}\sqrt{\frac{234{x}^{11}y}{26{x}^{7}y}}\hfill & \phantom{\rule{2em}{0ex}}\text{Combine numerator and denominator into one radical expression}.\hfill \\ \sqrt{9{x}^{4}}\hfill & \phantom{\rule{2em}{0ex}}\text{Simplify fraction}.\hfill \\ 3{x}^{2}\text{ }\hfill & \phantom{\rule{2em}{0ex}}\text{Simplify square root}.\hfill \end{array}$

### Try It

Simplify $\,\frac{\sqrt{9{a}^{5}{b}^{14}}}{\sqrt{3{a}^{4}{b}^{5}}}.$

${b}^{4}\sqrt{3ab}$

### 3.2.4 – Adding and Subtracting Square Roots

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.

### Example 6 – Adding Square Roots

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

We can rewrite $\,5\sqrt{12}\,$ as $\,5\sqrt{4\cdot3}.\,$ 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 $\,\sqrt{5}+6\sqrt{20}.$

$13\sqrt{5}$

### Example 7 – Subtracting Square Roots

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

Rewrite each term so they have equal radicands.

$\begin{array}{ccc}\hfill 20\sqrt{72{a}^{3}{b}^{4}c}& =& 20\sqrt{9}\sqrt{4}\sqrt{2}\sqrt{a}\sqrt{{a}^{2}}\sqrt{{\left({b}^{2}\right)}^{2}}\sqrt{c}\hfill \\ & =& 20\left(3\right)\left(2\right)|a|{b}^{2}\sqrt{2ac}\hfill \\ & =& 120|a|{b}^{2}\sqrt{2ac}\hfill \end{array}$
$\begin{array}{ccc}\hfill 14\sqrt{8{a}^{3}{b}^{4}c}& =& 14\sqrt{2}\sqrt{4}\sqrt{a}\sqrt{{a}^{2}}\sqrt{{\left({b}^{2}\right)}^{2}}\sqrt{c}\hfill \\ & =& 14\left(2\right)|a|{b}^{2}\sqrt{2ac}\hfill \\ & =& 28|a|{b}^{2}\sqrt{2ac}\hfill \end{array}$

Now the terms have the same radicand so we can subtract.

$120|a|{b}^{2}\sqrt{2ac}-28|a|{b}^{2}\sqrt{2ac}=92|a|{b}^{2}\sqrt{2ac}$

### Try It

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

$0$

### 3.2.5 – Rationalizing Denominators

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 $\,\frac{\sqrt{c}}{\sqrt{c}}.$

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 8 – Rationalizing a Denominator Containing a Single Term

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

The radical in the denominator is $\,\sqrt{10}.\,$ So multiply the fraction by $\,\frac{\sqrt{10}}{\sqrt{10}}.\,$ Then simplify.

$\begin{array}{l}\frac{2\sqrt{3}}{3\sqrt{10}}\cdot \frac{\sqrt{10}}{\sqrt{10}}\text{ }\\ \frac{2\sqrt{30}}{30}\text{ }\\ \frac{\sqrt{30}}{15}\end{array}$

### Try It

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

$6\sqrt{6}$

### 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 9 – Rationalizing a Denominator Containing Two Terms

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

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 $\,\frac{1-\sqrt{5}}{1-\sqrt{5}}.$

$\begin{array}{cc}\frac{4}{1+\sqrt{5}}\cdot \frac{1-\sqrt{5}}{1-\sqrt{5}}\hfill & \\ \frac{4-4\sqrt{5}}{-4}\hfill & \phantom{\rule{2em}{0ex}}\text{Use the distributive property}.\hfill \\ \sqrt{5}-1\hfill & \phantom{\rule{2em}{0ex}}\text{Simplify}.\hfill \end{array}$

### Try It

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

$14-7\sqrt{3}$

### Key Concepts

• The principal square root of a number $\,a\,$ is the nonnegative number that when multiplied by itself equals $\,a.\,$ See Example 1.
• If $\,a\,$ and $\,b\,$ are nonnegative, the square root of the product $\,ab\,$ is equal to the product of the square roots of $\,a\,$ and $\,b\,$ See Example 2 and Example 3.
• If $\,a\,$ and $\,b\,$ are nonnegative, the square root of the quotient $\,\frac{a}{b}\,$ is equal to the quotient of the square roots of $\,a\,$ and $\,b\,$ See Example 4 and Example 5.
• We can add and subtract radical expressions if they have the same radicand and the same index. See Example 6 and Example 7.
• Radical expressions written in simplest form do not contain a radical in the denominator. To eliminate the square root radical from the denominator, multiply both the numerator and the denominator by the conjugate of the denominator. See Example 8 and Example 9.

### Glossary

index
principal square root
the nonnegative square root of a number $\,a\,$ that, when multiplied by itself, equals $\,a$