Module 1.3 – Simplifying Radicals

Learning Objectives

In this section, you will:

- Evaluate square roots.
- Use the product rule to simplify square roots.
- Use the quotient rule to simplify square roots.
- Add and subtract square roots.
- Use rational roots.

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.

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.

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.

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: $\sqrt{a}$.

Q & A…

Does $\sqrt{25} = \pm 5$?

No. Although both $5^{2}$ and $(−5)^{2}$ 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.

- $\sqrt{100}$
- $\sqrt{\sqrt{16}}$
- $\sqrt{25+144}$
- $\sqrt{49} – \sqrt{81}$

Solution

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

Q & A…

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

- $\sqrt{225}$
- $\sqrt{\sqrt{81}}$
- $\sqrt{25-9}$
- $\sqrt{36} + \sqrt{121}$

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.

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.

- Factor any perfect squares from the radicand.
- Write the radical expression as a product of radical expressions.
- Simplify.

Example 2 Using the Product Rule to Simplify Square Roots

Simplify the radical expression.

- $\sqrt{300}$
- $\sqrt{162a^{5}b^{4}}$

Solution

- $\sqrt{300} = \sqrt{100 \cdot 3} = \sqrt{100} \cdot \sqrt{3} = 10 \sqrt{3}$
- $\sqrt{162a^{5}b^{4}} = \sqrt{81a^{4}b^{4} \cdot 2a} = \sqrt{81a^{4}b^{4}} \cdot \sqrt{2a} = 9a^{2}b^{2} \sqrt{2a}$

Try It

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

How To…

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

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

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

Simplify the radical expression.

\[

\sqrt{12} \cdot \sqrt{3}

\]

Solution

\[

\sqrt{12} \cdot \sqrt{3} = \sqrt{12 \cdot 3} = \sqrt{36} = 6

\]

Try It

Simplify $\sqrt{50x} \cdot \sqrt{2x}$.

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

\[

\sqrt{\dfrac{a}{b}} = \dfrac{\sqrt{a}}{\sqrt{b}}

\]

How To…

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

- Write the radical expression as the quotient of two radical expression.
- Simplify the numerator and denominator.

Example 4 Using the Quotient Rule to Simplify Square Roots

Simplify the radical expression.

\[

\sqrt{\dfrac{5}{36}}

\]

Solution

\[

\sqrt{\dfrac{5}{36}} = \dfrac{\sqrt{5}}{\sqrt{36}} = \dfrac{\sqrt{5}}{6}

\]

Try It

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

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

Simplify the radical expression.

\[

\dfrac{\sqrt{234x^{11}y}}{\sqrt{26x^{7}y}}

\]

Solution

\[

\dfrac{\sqrt{234x^{11}y}}{\sqrt{26x^{7}y}} = \sqrt{\dfrac{234x^{11}y}{26x^{7}y}} = \sqrt{9x^{4}} = 3x^{2}

\]

Try It

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

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.

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

Example 6 Adding Square Roots

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

Solution

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

\[

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

\]

Try It

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

Example 7 Subtracting Square Roots

Subtract $20\sqrt{72a^{3}b^{4}c} – 14\sqrt{8a^{3}b^{4}c}$, assuming all variables are positive.

Solution

Simplify the first term:

\[

20\sqrt{72a^{3}b^{4}c} = 20\sqrt{36a^{2}b^{4} \cdot 2ac} = 20(6ab^{2})\sqrt{2ac} = 120ab^{2}\sqrt{2ac}

\]

Simplify the second term:

\[

14\sqrt{8a^{3}b^{4}c} = 14\sqrt{4a^{2}b^{4} \cdot 2ac} = 14(2ab^{2})\sqrt{2ac} = 28ab^{2}\sqrt{2ac}

\]

Now the terms have the same variable and radical expressions, so we can subtract:

\[

20\sqrt{72a^{3}b^{4}c} – 14\sqrt{8a^{3}b^{4}c} = 120ab^{2}\sqrt{2ac} – 28ab^{2}\sqrt{2ac} = 92ab^{2}\sqrt{2ac}

\]

Try It

Subtract $3\sqrt{80x} – 4\sqrt{45x}$.

Using Rational Roots

Although square roots are the most common rational roots, we can also find cube roots, 4th roots, 5th roots, and more. Just as the square root function is the inverse of the squaring function, these roots are the inverse of their respective power functions. These functions can be useful when we need to determine the number that, when raised to a certain power, gives a certain number.

Understanding nth Roots

Suppose we know that $a^{3} = 8$. We want to find what number raised to the 3rd power is equal to 8. Since $2^{3} = 8$, we say that 2 is the cube root of 8.

The $n$th root of a is a number that, when raised to the $n$th power, gives $a$. For example, $-3$ is the 5th root of $-243$ because $(-3)^{5} = -243$. If $a$ is a real number with at least one $n$th root, then the principal $n$th root of $a$ is the number with the same sign as $a$ that, when raised to the $n$th power, equals $a$.

The principal $n$th root of $a$ is written as $\sqrt[n]{a}$ , where $n$ is a positive integer greater than or equal to 2. In the radical expression, $n$ is called the index of the radical.

If $a$ is a real number with at least one $n$th root, then the principal $n$th root of $a$, written as $\sqrt[n]{a}$ , is the number with the same sign as $a$ that, when raised to the $n$th power, equals $a$. The index of the radical is $n$.

Example 8 Simplifying nth Roots

Simplify each of the following:

- $\sqrt[5]{-32}$
- $\sqrt[4]{4} \cdot \sqrt[4]{1024}$
- $-\sqrt[3]{\dfrac{8x^{6}}{125}}$
- $8\sqrt[4]{3} – \sqrt[4]{48}$

Solution

- $\sqrt[5]{-32} = -2$ because $(-2)^{5} = -32$.
- First, express the product as a single radical expression:

\[

\sqrt[4]{4} \cdot \sqrt[4]{1024} = \sqrt[4]{4096} = 8 \text{ because } 8^{4} = 4096.

\] - $-\sqrt[3]{\dfrac{8x^{6}}{125}} = -\dfrac{\sqrt[3]{8x^{6}}}{\sqrt[3]{125}} = -\dfrac{2x^{2}}{5}$
- $8\sqrt[4]{3} – \sqrt[4]{48} = 8\sqrt[4]{3} – 2\sqrt[4]{3} = 6\sqrt[4]{3}$

Try It

Simplify.

- $\sqrt[3]{-216}$
- $\dfrac{3\sqrt[4]{80}}{\sqrt[4]{5}}$
- $6\sqrt[3]{9000} + 7\sqrt[3]{576}$

## Using Rational Exponents

Radical expressions can also be written without using the radical symbol. We can use rational (fractional) exponents. The index must be a positive integer. If the index $n$ is even, then $a$ cannot be negative.

\[

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

\]

We can also have rational exponents with numerators other than 1. In these cases, the exponent must be a fraction in lowest terms. We raise the base to a power and take an $n$th root. The numerator tells us the power and the denominator tells us the root.

\[

a^{\frac{m}{n}} = (\sqrt[n]{a})^{m} = \sqrt[n]{a^{m}}

\]

All of the properties of exponents that we learned for integer exponents also hold for rational exponents.

Rational exponents are another way to express principal $n$th roots. The general form for converting between a radical expression with a radical symbol and one with a rational exponent is

\[

a^{\frac{m}{n}} = (\sqrt[n]{a})^{m} = \sqrt[n]{a^{m}}.

\]

How To…

Given an expression with a rational exponent, write the expression as a radical.

- Determine the power by looking at the numerator of the exponent.
- Determine the root by looking at the denominator of the exponent.
- Using the base as the radicand, raise the radicand to the power and use the root as the index.

Example 9 Writing Rational Exponents as Radicals

Write $343^{\frac{2}{3}}$ as a radical. Simplify.

Solution The 2 tells us the power and the 3 tells us the root.

\[

343^{\frac{2}{3}} = (\sqrt[3]{343})^{2} = \sqrt[3]{343^{2}}

\]

We know that $\sqrt[3]{343} = 7$ because $7^{3} = 343$. Because the cube root is easy to find, it is easiest to find the cube root before squaring for this problem. In general, it is easier to find the root first and then raise it to a power.

\[

343^{\frac{2}{3}} = (\sqrt[3]{343})^{2} = 7^{2} = 49

\]

Try It

Write $9^{\frac{5}{2}}$ as a radical. Simplify.