1.4.2: Simplifying Radical Expressions

Learning Objectives

By the end of this section, you will be able to:

• Use the Product Property to simplify radical expressions
• Use the Quotient Property to simplify radical expressions

Be Prepared

Before you get started, take this readiness quiz.

1. Simplify $\dfrac{x^{9}}{x^{4}}$.
2. Simplify $\dfrac{y^{3}}{y^{11}}$.
3. Simplify $\left(n^{2}\right)^{6}$.

We will simplify radical expressions in a way similar to how we simplified fractions. A fraction is simplified if there are no common factors in the numerator and denominator. To simplify a fraction, we look for any common factors in the numerator and denominator. A radical expression, ⁿ√ a , is considered simplified if it has no factors of mⁿ. So, to simplify a radical expression, we look for any factors in the radicand that are powers of the index.

Use the Product Property to Simplify Radical Expressions

We will simplify radical expressions in a way similar to how we simplified fractions. A fraction is simplified if there are no common factors in the numerator and denominator. To simplify a fraction, we look for any common factors in the numerator and denominator.

A radical expression, $\sqrt{a}$, is considered simplified if it has no factors of the form $m^{2}$. So, to simplify a radical expression, we look for any factors in the radicand that are squares.

Definition $\PageIndex{1}$

For non-negative integers $a$ and $m$,

$\sqrt{a}$ is considered simplified if $a$ has no factors of the form $m^{2}$.

For example, $\sqrt{5}$ is considered simplified because there are no perfect square factors in $5$. But $\sqrt{12}$ is not simplified because $12$ has a perfect square factor of $4$.

To simplify radical expressions, we will also use some properties of roots. The properties we will use to simplify radical expressions are similar to the properties of exponents. We know that

$$(a b)^{n}=a^{n} b^{n}.$$

The corresponding of Product Property of Roots says that

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

To see why this is true we note that

$$(\sqrt{ab})^2=ab$$

since $\sqrt{ab}$ is the non-negative quantity you square to get $ab$ by definition.

Also,

$$(\sqrt{a} \cdot \sqrt{b})^2=(\sqrt{a})^2 \cdot (\sqrt{b})^2=ab,$$

where the first equality follows from the product property of exponents and the second by the definition of the square root (as above).

So, the left and the right hand sides, being both non-negative, are square roots of $ab$, and therefore are equal.

As mentioned in the last section, due to this property, it is written that $\sqrt{a}=a^{\frac12}$ and the properties of exponents can be shown to be extended to the exponents obtained this way.

Fact $\PageIndex{2}$

If $\sqrt{a}$ and $\sqrt{b}$ are real numbers, and $n\geq 2$ is an integer, then

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

Note that you can also read the equality: $\quad \sqrt{a} \cdot \sqrt{b}=\sqrt{a b}$.

We use the Product Property of Roots to remove all perfect square factors from a square root.

Example $\PageIndex{3}$

Simplify $\sqrt{98}$.

Solution

Find the largest factor in the radicand that is a perfect power of the index.	We see that $49$ is the largest factor of $98$ that has a power of $2$.	$\sqrt{98}$
Rewrite the radicand as a product of two factors, using that factor.	In other words $49$ is the largest perfect square factor of $98$. $98 = 49\cdot 2$ Always write the perfect square factor first.	$\sqrt{49\cdot 2}$
Use the product rule to rewrite the radical as the product of two radicals.		$\sqrt{49} \cdot \sqrt{2}$
Simplify the root of the perfect power.		$7\sqrt{2}$

Try It $\PageIndex{4}$

Simplify $\sqrt{48}$.

Answer

$4 \sqrt{3}$

Try It $\PageIndex{5}$

Simplify $\sqrt{45}$.

Answer

$3 \sqrt{5}$

Notice in the previous example that the simplified form of $\sqrt{98}$ is $7\sqrt{2}$, which is the product of an integer and a square root. We always write the integer in front of the square root.

Be careful to write your integer so that it is not confused with the index (which we will discuss later). The expression $7\sqrt{2}$ is very different from $\sqrt[7]{2}$.

Simplify a Radical Expression Using the Product Property

1. Find the largest factor in the radicand that is a perfect power of the index. Rewrite the radicand as a product of two factors, using that factor.
2. Use the product rule to rewrite the radical as the product of two radicals.
3. Simplify the root of the perfect power.

We will apply this method in the next example. It may be helpful to have a table of perfect squares.

Example $\PageIndex{6}$

Simplify $\sqrt{500}$.

Solution

$\sqrt{500}$

Rewrite the radicand as a product using the largest perfect square factor.

$\sqrt{100 \cdot 5}$

Rewrite the radical as the product of two radicals.

$\sqrt{100} \cdot \sqrt{5}$

Simplify.

$10\sqrt{5}$

Try It $\PageIndex{7}$

Simplify $\sqrt{288}$.

Answer

$12\sqrt{2}$

Try It $\PageIndex{8}$

Simplify $\sqrt{432}$.

Answer

$12\sqrt{3}$

The next example is much like the previous examples, but with variables. Don’t forget to use the absolute value signs when taking an even root of an expression with a variable in the radical.

Example $\PageIndex{9}$

Simplify $\sqrt{x^{3}}$.

Solution

$\sqrt{x^{3}}$

Rewrite the radicand as a product using the largest perfect square factor.

$\sqrt{x^{2} \cdot x}$

Rewrite the radical as the product of two radicals.

$\sqrt{x^{2}} \cdot \sqrt{x}$

Simplify.

$|x| \sqrt{x}$

Try It $\PageIndex{10}$

Simplify $\sqrt{b^{5}}$.

Answer

$b^{2} \sqrt{b}$

Try It $\PageIndex{11}$

Simplify $\sqrt{p^{9}}$.

Answer

$p^{4} \sqrt{p}$

We follow the same procedure when there is a coefficient in the radicand. In the next example, both the constant and the variable have perfect square factors.

Example $\PageIndex{12}$

Simplify $\sqrt{72 n^{7}}$.

Solution

$\sqrt{72 n^{7}}$

Rewrite the radicand as a product using the largest perfect square factor.

$\sqrt{36 n^{6} \cdot 2 n}$

Rewrite the radical as the product of two radicals.

$\sqrt{36 n^{6}} \cdot \sqrt{2 n}$

Simplify.

$6\left|n^{3}\right| \sqrt{2 n}$

Try It $\PageIndex{13}$

Simplify $\sqrt{32 y^{5}}$.

Answer

$4 y^{2} \sqrt{2 y}$

Try It $\PageIndex{14}$

Simplify $\sqrt{75 a^{9}}$.

Answer

$5 a^{4} \sqrt{3 a}$

In the next example, we continue to use the same methods even though there are more than one variable under the radical.

Example $\PageIndex{15}$

Simplify $\sqrt{63 u^{3} v^{5}}$.

Answer

$\sqrt{63 u^{3} v^{5}}$

Rewrite the radicand as a product using the largest perfect square factor.

$\sqrt{9 u^{2} v^{4} \cdot 7 u v}$

Rewrite the radical as the product of two radicals.

$\sqrt{9 u^{2} v^{4}} \cdot \sqrt{7 u v}$

Rewrite the first radicand as $\left(3 u v^{2}\right)^{2}$.

$\sqrt{\left(3 u v^{2}\right)^{2}} \cdot \sqrt{7 u v}$

Simplify.

$3|u| v^{2} \sqrt{7 u v}$

Try It $\PageIndex{16}$

Simplify $\sqrt{98 a^{7} b^{5}}$.

Answer

$7\left|a^{3}\right| b^{2} \sqrt{2 a b}$

Try It $\PageIndex{17}$

Simplify $\sqrt{180 m^{9} n^{11}}$.

Answer

$6 m^{4}\left|n^{5}\right| \sqrt{5 m n}$

Use the Quotient Property to Simplify Radical Expressions

Whenever you have to simplify a radical expression, the first step you should take is to determine whether the radicand is a perfect square. If not, check the numerator and denominator for any common factors, and remove them. You may find a fraction in which both the numerator and the denominator are perfect powers of the index.

Example $\PageIndex{18}$

Simplify $\sqrt{\dfrac{45}{80}}$.

Solution

$\sqrt{\dfrac{45}{80}}$

Simplify inside the radical first. Rewrite showing the common factors of the numerator and denominator.

$\sqrt{\dfrac{5 \cdot 9}{5 \cdot 16}}$

Simplify the fraction by removing common factors.

$\sqrt{\dfrac{9}{16}}$

Simplify. Note $\left(\dfrac{3}{4}\right)^{2}=\dfrac{9}{16}$.

$\dfrac{3}{4}$

Try It $\PageIndex{19}$

Simplify $\sqrt{\dfrac{75}{48}}$.

Answer

$\dfrac{5}{4}$

Try It $\PageIndex{20}$

Simplify $\sqrt{\dfrac{98}{162}}$.

Answer

$\dfrac{7}{9}$

In the last example, our first step was to simplify the fraction under the radical by removing common factors. In the next example we will use the Quotient Property to simplify under the radical. We divide the like bases by subtracting their exponents,

$\dfrac{a^{m}}{a^{n}}=a^{m-n}, \quad a \neq 0$

Example $\PageIndex{21}$

Simplify $\sqrt{\dfrac{m^{6}}{m^{4}}}$.

Solution

$\sqrt{\dfrac{m^{6}}{m^{4}}}$

Simplify the fraction inside the radical first. Divide the like bases by subtracting the exponents.

$\sqrt{m^{2}}$

Simplify.

$|m|$

Try It $\PageIndex{22}$

Simplify $\sqrt{\dfrac{a^{8}}{a^{6}}}$.

Answer

$|a|$

Try It $\PageIndex{23}$

Simplify $\sqrt{\dfrac{x^{14}}{x^{10}}}$.

Answer

$x^{2}$

Remember the Quotient to a Power Property? It said we could raise a fraction to a power by raising the numerator and denominator to the power separately.

$\left(\dfrac{a}{b}\right)^{m}=\dfrac{a^{m}}{b^{m}}, b \neq 0$

Quotient Property of Radical Expressions

If $\sqrt{a}$ and $\sqrt{b}$ are real numbers, $b \neq 0$, and for any integer $n \geq 2$ then,

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

Example $\PageIndex{24}$

Simplify $\sqrt{\dfrac{27 m^{3}}{196}}$.

Solution

Simplify the fraction in the radicand, if possible.

$\dfrac{27 m^{3}}{196}$ cannot be simplified.

$\sqrt{\dfrac{27 m^{3}}{196}}$

Use the Quotient Property to rewrite the radical as the quotient of two radicals.

We rewrite $\sqrt{\dfrac{27 m^{3}}{196}}$ as the quotient of $\sqrt{27 m^{3}}$ and $\sqrt{196}$.

$\dfrac{\sqrt{27 m^{3}}}{\sqrt{196}}$

Simplify the radicals in the numerator and the denominator.

$9m^{2}$ and $196$ are perfect squares.

$\dfrac{\sqrt{9 m^{2}} \cdot \sqrt{3 m}}{\sqrt{196}}$

$\dfrac{3 m \sqrt{3 m}}{14}$

Try It $\PageIndex{25}$

Simplify $\sqrt{\dfrac{24 p^{3}}{49}}$.

Answer

$\dfrac{2|p| \sqrt{6 p}}{7}$

Try It $\PageIndex{26}$

Simplify $\sqrt{\dfrac{48 x^{5}}{100}}$.

Answer

$\dfrac{2 x^{2} \sqrt{3 x}}{5}$

Simplify a Square Root Using the Quotient Property

1. Simplify the fraction in the radicand, if possible.
2. Use the Quotient Property to rewrite the radical as the quotient of two radicals.
3. Simplify the radicals in the numerator and the denominator.

Example $\PageIndex{27}$

Simplify $\sqrt{\dfrac{45 x^{5}}{y^{4}}}$.

Solution

$\sqrt{\dfrac{45 x^{5}}{y^{4}}}$

We cannot simplify the fraction in the radicand. Rewrite using the Quotient Property.

$\dfrac{\sqrt{45 x^{5}}}{\sqrt{y^{4}}}$

Simplify the radicals in the numerator and the denominator.

$\dfrac{\sqrt{9 x^{4}} \cdot \sqrt{5 x}}{y^{2}}$

Simplify.

$\dfrac{3 x^{2} \sqrt{5 x}}{y^{2}}$

Try It $\PageIndex{28}$

Simplify $\sqrt{\dfrac{80 m^{3}}{n^{6}}}$.

Answer

$\dfrac{4|m| \sqrt{5 m}}{\left|n^{3}\right|}$

Try It $\PageIndex{29}$

Simplify $\sqrt{\dfrac{54 u^{7}}{v^{8}}}$.

Answer

$\dfrac{3 u^{3} \sqrt{6 u}}{v^{4}}$

Be sure to simplify the fraction in the radicand first, if possible.

Example $\PageIndex{30}$

Simplify $\sqrt{\dfrac{18 p^{5} q^{7}}{32 p q^{2}}}$.

Solution

$\sqrt{\dfrac{18 p^{5} q^{7}}{32 p q^{2}}}$

Simplify the fraction in the radicand, if possible.

$\sqrt{\dfrac{9 p^{4} q^{5}}{16}}$

Rewrite using the Quotient Property.

$\dfrac{\sqrt{9 p^{4} q^{5}}}{\sqrt{16}}$

Simplify the radicals in the numerator and the denominator.

$\dfrac{\sqrt{9 p^{4} q^{4}} \cdot \sqrt{q}}{4}$

Simplify.

$\dfrac{3 p^{2} q^{2} \sqrt{q}}{4}$

Try It $\PageIndex{31}$

Simplify $\sqrt{\dfrac{50 x^{5} y^{3}}{72 x^{4} y}}$.

Answer

$\dfrac{5|y| \sqrt{x}}{6}$

Try It $\PageIndex{32}$

Simplify $\sqrt{\dfrac{48 m^{7} n^{2}}{100 m^{5} n^{8}}}$.

Answer

$\dfrac{2|m| \sqrt{3}}{5\left|n^{3}\right|}$

In the next example, there is nothing to simplify in the denominators. Since the index on the radicals is the same, we can use the Quotient Property again, to combine them into one radical. We will then look to see if we can simplify the expression.

Example $\PageIndex{33}$

Simplify $\dfrac{\sqrt{48 a^{7}}}{\sqrt{3 a}}$.

Solution

$\dfrac{\sqrt{48 a^{7}}}{\sqrt{3 a}}$

The denominator cannot be simplified, so use the Quotient Property to write as one radical.

$\sqrt{\dfrac{48 a^{7}}{3 a}}$

Simplify the fraction under the radical.

$\sqrt{16 a^{6}}$

Simplify.

$4\left|a^{3}\right|$

Try It $\PageIndex{34}$

Simplify $\dfrac{\sqrt{98 z^{5}}}{\sqrt{2 z}}$.

Answer

$7z^{2}$

Try It $\PageIndex{35}$

Simplify $\dfrac{\sqrt{128 m^{9}}}{\sqrt{2 m}}$.

Answer

$8m^{4}$

Writing Exercises $\PageIndex{36}$

1. What is the goal in simplifying a radical expression? 
2. How can you recognize if a radical expression is simplified? 

Exit Problem $\PageIndex{37}$

Simplify $-4x^3y\sqrt{32x^6y^9}$.

Key Concepts

• Simplified Radical Expression
  • For real numbers $a, m$ and $n≥2$
    $\sqrt{a}$ is considered simplified if $a$ has no factors of $m^{2}$
• Product Property of $n^{th}$ Roots
  • For any real numbers, $\sqrt{a}$ and $\sqrt{b}$, and for any integer $n≥2$
    $\sqrt{a b}=\sqrt{a} \sqrt{b}$ and $\sqrt{a} \sqrt{b}=\sqrt{a b}$
• How to simplify a radical expression using the Product Property
  1. Find the largest factor in the radicand that is a perfect power of the index.
     Rewrite the radicand as a product of two factors, using that factor.
  2. Use the product rule to rewrite the radical as the product of two radicals.
  3. Simplify the root of the perfect power.
• Quotient Property of Radical Expressions
  • If $\sqrt{a}$ and $\sqrt{b}$ are real numbers, $b≠0$, and for any integer $n≥2$ then, $\sqrt{\dfrac{a}{b}}=\dfrac{\sqrt{a}}{\sqrt{b}}$.
• How to simplify a radical expression using the Quotient Property.
  1. Simplify the fraction in the radicand, if possible.
  2. Use the Quotient Property to rewrite the radical as the quotient of two radicals.
  3. Simplify the radicals in the numerator and the denominator.