8.2 Simplify Radical Expressions

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, $\sqrt[n]{a},$ is considered simplified if it has no factors of $m^{n} .$ So, to simplify a radical expression, we look for any factors in the radicand that are powers of the index.

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

Similarly, $the cube root of 4$ is simplified because there are no perfect cube factors in 4. But $the cube root of 24$ is not simplified because 24 has a perfect cube factor of 8.

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 $\sqrt[n]{a b} = \sqrt[n]{a} \cdot \sqrt[n]{b} .$

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

Notice in the previous example that the simplified form of $the square root of 98$ is $7 the square root of 2,$ 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. The expression $7 the square root of 2$ is very different from $the 7th root of 2.$

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

Example 2

Simplify: ⓐ $the square root of 500$ ⓑ $the cube root of 16$ ⓒ $the 4th root of 243.$

ⓐ

	$the square root of 500$
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.	$10 the square root of 5$

ⓑ

	$the cube root of 16$
Rewrite the radicand as a product using the greatest perfect cube factor. $2 cubed equals 8$	$\sqrt[3]{8 \cdot 2}$
Rewrite the radical as the product of two radicals.	$\sqrt[3]{8} \cdot \sqrt[3]{2^{}}$
Simplify.	$2 the cube root of 2$

ⓒ

	$the 4th root of 243$
Rewrite the radicand as a product using the greatest perfect fourth power factor. $3^{4} = 81$	$\sqrt[4]{81 \cdot 3^{}}$
Rewrite the radical as the product of two radicals.	$\sqrt[4]{81} \cdot \sqrt[4]{3^{}}$
Simplify.	$3 the 4th root of 3$

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 3

Simplify: ⓐ $\sqrt{x^{3}}$ ⓑ $\sqrt[3]{x^{4}}$ ⓒ $\sqrt[4]{x^{7}} .$

ⓐ

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

ⓑ

	$\sqrt[3]{x^{4}}$
Rewrite the radicand as a product using the largest perfect cube factor.	$\sqrt[3]{x^{3} \cdot x} .$
Rewrite the radical as the product of two radicals.	$\sqrt[3]{x^{3}} \cdot \sqrt[3]{x}$
Simplify.	$x \sqrt[3]{x}$

ⓒ

	$\sqrt[4]{x^{7}}$
Rewrite the radicand as a product using the greatest perfect fourth power factor.	$\sqrt[4]{x^{4} \cdot x^{3}}$
Rewrite the radical as the product of two radicals.	$\sqrt[4]{x^{4}} \cdot \sqrt[4]{x^{3}}$
Simplify.	$vertical bar x vertical bar the 4th root of x cubed$

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 4

Simplify: ⓐ $\sqrt{72 n^{7}}$ ⓑ $\sqrt[3]{24 x^{7}}$ ⓒ $\sqrt[4]{80 y^{14}} .$

ⓐ

	$\sqrt{72 n^{7}}$
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.	$6 vertical bar n cubed vertical bar the square root of 2 n$

ⓑ

	$\sqrt[3]{24 x^{7}}$
Rewrite the radicand as a product using perfect cube factors.	$\sqrt[3]{8 x^{6} \cdot 3 x}$
Rewrite the radical as the product of two radicals.	$\sqrt[3]{8 x^{6}} \cdot \sqrt[3]{3^{} x}$
Rewrite the first radicand as $open parenthesis 2 x squared close parenthesis cubed.$	$\sqrt[3]{{(2^{} x^{2})}^{3}} \cdot \sqrt[3]{3^{} x}$
Simplify.	$2 x squared the cube root of 3 x$

ⓒ

	$\sqrt[4]{80 y^{14}}$
Rewrite the radicand as a product using perfect fourth power factors.	$\sqrt[4]{16 y^{12} \cdot 5^{} y^{2}}$
Rewrite the radical as the product of two radicals.	$\sqrt[4]{16 y^{12}} \cdot \sqrt[4]{5 y^{2}^{}}$
Rewrite the first radicand as $open parenthesis 2 y cubed close parenthesis to the power 4.$	$\sqrt[4]{{(2^{} y^{3})}^{4}} \cdot \sqrt[4]{5 y^{2}^{}}$
Simplify.	$2 vertical bar y cubed vertical bar the 4th root of 5 y squared$

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

Example 5

Simplify: ⓐ $\sqrt{63 u^{3} v^{5}}$ ⓑ $\sqrt[3]{40 x^{4} y^{5}}$ ⓒ $\sqrt[4]{48 x^{4} y^{7}} .$

ⓐ

	$\sqrt{63 u^{3} v^{5}}$
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 $open parenthesis 3 u v squared close parenthesis squared.$	$\sqrt{{(3 u v^{2})}^{2}} \cdot \sqrt{7 u v}$
Simplify.	$3 vertical bar u vertical bar v squared the square root of 7 u v$

ⓑ

	$\sqrt[3]{40 x^{4} y^{5}}$
Rewrite the radicand as a product using the largest perfect cube factor.	$\sqrt[3]{8 x^{3} y^{3} \cdot 5 x y^{2}}$
Rewrite the radical as the product of two radicals.	$\sqrt[3]{8 x^{3} y^{3}} \cdot \sqrt[3]{5 x y^{2}}$
Rewrite the first radicand as $open parenthesis 2 x y close parenthesis cubed.$	$\sqrt[3]{{(2 x y)}^{3}} \cdot \sqrt[3]{5 x y^{2}}$
Simplify.	$2 x y the cube root of 5 x y squared$

ⓒ

	$\sqrt[4]{48 x^{4} y^{7}}$
Rewrite the radicand as a product using the largest perfect fourth power factor.	$\sqrt[4]{16 x^{4} y^{4} \cdot 3 y^{3}}$
Rewrite the radical as the product of two radicals.	$\sqrt[4]{16 x^{4} y^{4}} \cdot \sqrt[4]{3 y^{3}}$
Rewrite the first radicand as $open parenthesis 2 x y close parenthesis to the power 4.$	$\sqrt[4]{{(2 x y)}^{4}} \cdot \sqrt[4]{3 y^{3}}$
Simplify.	$2 vertical bar x y vertical bar the 4th root of 3 y cubed$

Example 6

Simplify: ⓐ $the cube root of −27$ ⓑ $the 4th root of −16.$

ⓐ

	$the cube root of −27$
Rewrite the radicand as a product using perfect cube factors.	$the cube root of open parenthesis −3 close parenthesis cubed$
Take the cube root.	$−3$

ⓑ

	$the 4th root of −16$
There is no real number $n$ where $n^{4} = −16 .$	Not a real number.

We have seen how to use the order of operations to simplify some expressions with radicals. In the next example, we have the sum of an integer and a square root. We simplify the square root but cannot add the resulting expression to the integer since one term contains a radical and the other does not. The next example also includes a fraction with a radical in the numerator. Remember that in order to simplify a fraction you need a common factor in the numerator and denominator.

Example 7

Simplify: ⓐ $3 + \sqrt{32}$ ⓑ $\frac{4 - \sqrt{48}}{2} .$

ⓐ

	$3 + \sqrt{32}$
Rewrite the radicand as a product using the largest perfect square factor.	$3 + \sqrt{16 \cdot 2}$
Rewrite the radical as the product of two radicals.	$3 + \sqrt{16} \cdot \sqrt{2}$
Simplify.	$3 + 4 \sqrt{2}$

The terms cannot be added as one has a radical and the other does not. Trying to add an integer and a radical is like trying to add an integer and a variable. They are not like terms!

ⓑ

	$\frac{4 - \sqrt{48}}{2}$
Rewrite the radicand as a product using the largest perfect square factor.	$\frac{4 - \sqrt{16 \cdot 3}}{2}$
Rewrite the radical as the product of two radicals.	$\frac{4 - \sqrt{16} \cdot \sqrt{3}}{2}$
Simplify.	$\frac{4 - 4 \sqrt{3}}{2}$
Factor the common factor from the numerator.	$\frac{4 (1 - \sqrt{3})}{2}$
Remove the common factor, 2, from the numerator and denominator.	$\frac{2 \cdot 2 (1 - \sqrt{3})}{2}$
Simplify.	$2 (1 - \sqrt{3})$

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 power of the index. 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 8

Simplify: ⓐ $\sqrt{\frac{45}{80}}$ ⓑ $\sqrt[3]{\frac{16}{54}}$ ⓒ $\sqrt[4]{\frac{5}{80}} .$

ⓐ

	$\sqrt{\frac{45}{80}}$
Simplify inside the radical first.
Rewrite showing the common factors of the numerator and denominator.	$\sqrt{\frac{5 \cdot 9}{5 \cdot 16}}$
Simplify the fraction by removing common factors.	$\sqrt{\frac{9}{16}}$
Simplify. Note ${(\frac{3}{4})}^{2} = \frac{9}{16} .$	$\frac{3}{4}$

ⓑ

	$\sqrt[3]{\frac{16}{54}}$
Simplify inside the radical first.
Rewrite showing the common factors of the numerator and denominator.	$\sqrt[3]{\frac{2 \cdot 8}{2 \cdot 27}}$
Simplify the fraction by removing common factors.	$\sqrt[3]{\frac{8}{27}}$
Simplify. Note ${(\frac{2}{3})}^{3} = \frac{8}{27} .$	$\frac{2}{3}$

ⓒ

	$\sqrt[4]{\frac{5}{80}}$
Simplify inside the radical first.
Rewrite showing the common factors of the numerator and denominator.	$\sqrt[4]{\frac{5 \cdot 1}{5 \cdot 16}}$
Simplify the fraction by removing common factors.	$\sqrt[4]{\frac{1}{16}}$
Simplify. Note ${(\frac{1}{2})}^{4} = \frac{1}{16} .$	$\frac{1}{2}$

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,

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

Example 9

Simplify: ⓐ $\sqrt{\frac{m^{6}}{m^{4}}}$ ⓑ $\sqrt[3]{\frac{a^{8}}{a^{5}}}$ ⓒ $\sqrt[4]{\frac{a^{10}}{a^{2}}} .$

ⓐ

	$\sqrt{\frac{m^{6}}{m^{4}}}$
Simplify the fraction inside the radical first.
Divide the like bases by subtracting the exponents.	$\sqrt{m^{2}}$
Simplify.	$\| m \|$

ⓑ

	$\sqrt[3]{\frac{a^{8}}{a^{5}}}$
Use the Quotient Property of exponents to simplify the fraction under the radical first.	$\sqrt[3]{a^{3}}$
Simplify.	$a$

ⓒ

	$\sqrt[4]{\frac{a^{10}}{a^{2}}}$
Use the Quotient Property of exponents to simplify the fraction under the radical first.	$\sqrt[4]{a^{8}}$
Rewrite the radicand using perfect fourth power factors.	$\sqrt[4]{{(a^{2})}^{4}}$
Simplify.	$a^{2}$

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.

${(\frac{a}{b})}^{m} = \frac{a^{m}}{b^{m}}, b \neq 0$

We can use a similar property to simplify a root of a fraction. After removing all common factors from the numerator and denominator, if the fraction is not a perfect power of the index, we simplify the numerator and denominator separately.

Example 11

Simplify: ⓐ $\sqrt{\frac{45 x^{5}}{y^{4}}}$ ⓑ $\sqrt[3]{\frac{24 x^{7}}{y^{3}}}$ ⓒ $\sqrt[4]{\frac{48 x^{10}}{y^{8}}} .$

ⓐ

	$\sqrt{\frac{45 x^{5}}{y^{4}}}$
We cannot simplify the fraction in the radicand. Rewrite using the Quotient Property.	$\frac{\sqrt{45 x^{5}}}{\sqrt{y^{4}}}$
Simplify the radicals in the numerator and the denominator.	$\frac{\sqrt{9 x^{4}} \cdot \sqrt{5 x}}{y^{2}}$
Simplify.	$\frac{3 x^{2} \sqrt{5 x}}{y^{2}}$

ⓑ

	$\sqrt[3]{\frac{24 x^{7}}{y^{3}}}$
The fraction in the radicand cannot be simplified. Use the Quotient Property to write as two radicals.	$\frac{\sqrt[3]{24 x^{7}}}{\sqrt[3]{y^{3}}}$
Rewrite each radicand as a product using perfect cube factors.	$\frac{\sqrt[3]{8 x^{6} \cdot 3 x}}{\sqrt[3]{y^{3}}}$
Rewrite the numerator as the product of two radicals.	$\frac{\sqrt[3]{{(2 x^{2})}^{3}} \cdot \sqrt[3]{3 x}}{\sqrt[3]{y^{3}}}$
Simplify.	$\frac{2 x^{2} \sqrt[3]{3 x}}{y}$

ⓒ

	$\sqrt[4]{\frac{48 x^{10}}{y^{8}}}$
The fraction in the radicand cannot be simplified.	$\frac{\sqrt[4]{48 x^{10}}}{\sqrt[4]{y^{8}}}$
Use the Quotient Property to write as two radicals. Rewrite each radicand as a product using perfect fourth power factors.	$\frac{\sqrt[4]{16 x^{8} \cdot 3 x^{2}}}{\sqrt[4]{y^{8}}}$
Rewrite the numerator as the product of two radicals.	$\frac{\sqrt[4]{{(2 x^{2})}^{4}} \cdot \sqrt[4]{3 x^{2}}}{\sqrt[4]{{(y^{2})}^{4}}}$
Simplify.	$\frac{2 x^{2} \sqrt[4]{3 x^{2}}}{y^{2}}$

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

Example 12

Simplify: ⓐ $\sqrt{\frac{18 p^{5} q^{7}}{32 p q^{2}}}$ ⓑ $\sqrt[3]{\frac{16 x^{5} y^{7}}{54 x^{2} y^{2}}}$ ⓒ $\sqrt[4]{\frac{5 a^{8} b^{6}}{80 a^{3} b^{2}}} .$

ⓐ

	$\sqrt{\frac{18 p^{5} q^{7}}{32 p q^{2}}}$
Simplify the fraction in the radicand, if possible.	$\sqrt{\frac{9 p^{4} q^{5}}{16}}$
Rewrite using the Quotient Property.	$\frac{\sqrt{9 p^{4} q^{5}}}{\sqrt{16}}$
Simplify the radicals in the numerator and the denominator.	$\frac{\sqrt{9 p^{4} q^{4}} \cdot \sqrt{q}}{4}$
Simplify.	$\frac{3 p^{2} q^{2} \sqrt{q}}{4}$

ⓑ

	$\sqrt[3]{\frac{16 x^{5} y^{7}}{54 x^{2} y^{2}}}$
Simplify the fraction in the radicand, if possible.	$\sqrt[3]{\frac{8 x^{3} y^{5}}{27}}$
Rewrite using the Quotient Property.	$\frac{\sqrt[3]{8 x^{3} y^{5}}}{\sqrt[3]{27}}$
Simplify the radicals in the numerator and the denominator.	$\frac{\sqrt[3]{8 x^{3} y^{3}} \cdot \sqrt[3]{y^{2}}}{\sqrt[3]{27}}$
Simplify.	$\frac{2 x y \sqrt[3]{y^{2}}}{3}$

ⓒ

	$\sqrt[4]{\frac{5 a^{8} b^{6}}{80 a^{3} b^{2}}}$
Simplify the fraction in the radicand, if possible.	$\sqrt[4]{\frac{a^{5} b^{4}}{16}}$
Rewrite using the Quotient Property.	$\frac{\sqrt[4]{a^{5} b^{4}}}{\sqrt[4]{16}}$
Simplify the radicals in the numerator and the denominator.	$\frac{\sqrt[4]{a^{4} b^{4}} \cdot \sqrt[4]{a}}{\sqrt[4]{16}}$
Simplify.	$\frac{\| a b \| \sqrt[4]{a}}{2}$

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 13

Simplify: ⓐ $\frac{\sqrt{48 a^{7}}}{\sqrt{3 a}}$ ⓑ $\frac{\sqrt[3]{−108}}{\sqrt[3]{2}}$ ⓒ $\frac{\sqrt[4]{96 x^{7}}}{\sqrt[4]{3 x^{2}}} .$

ⓐ

	$\frac{\sqrt{48 a^{7}}}{\sqrt{3 a}}$
The denominator cannot be simplified, so use the Quotient Property to write as one radical.	$\sqrt{\frac{48 a^{7}}{3 a}}$
Simplify the fraction under the radical.	$\sqrt{16 a^{6}}$
Simplify.	$4 vertical bar a cubed vertical bar$

ⓑ

	$\frac{\sqrt[3]{−108}}{\sqrt[3]{2}}$
The denominator cannot be simplified, so use the Quotient Property to write as one radical.	$\sqrt[3]{\frac{−108}{2}}$
Simplify the fraction under the radical.	$the cube root of −54$
Rewrite the radicand as a product using perfect cube factors.	$\sqrt[3]{{(−3)}^{3} \cdot 2}$
Rewrite the radical as the product of two radicals.	$\sqrt[3]{{(−3)}^{3}} \cdot \sqrt[3]{2}$
Simplify.	$−3 the cube root of 2$

ⓒ

	$\frac{\sqrt[4]{96 x^{7}}}{\sqrt[4]{3 x^{2}}}$
The denominator cannot be simplified, so use the Quotient Property to write as one radical.	$\sqrt[4]{\frac{96 x^{7}}{3 x^{2}}}$
Simplify the fraction under the radical.	$\sqrt[4]{32 x^{5}}$
Rewrite the radicand as a product using perfect fourth power factors.	$\sqrt[4]{16 x^{4}} \cdot \sqrt[4]{2 x}$
Rewrite the radical as the product of two radicals.	$\sqrt[4]{{(2 x)}^{4}} \cdot \sqrt[4]{2 x}$
Simplify.	$2 vertical bar x vertical bar the 4th root of 2 x$

Key Concepts

Simplified Radical Expression
- For real numbers a, m and $n greater than or equal to 2$
  $\sqrt[n]{a}$ is considered simplified if a has no factors of $m^{n}$
Product Property of n^th Roots
- For any real numbers, $\sqrt[n]{a}$ and $\sqrt[n]{b},$ and for any integer $n greater than or equal to 2$
  $\sqrt[n]{a b} = \sqrt[n]{a} \cdot \sqrt[n]{b}$ and $\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{a b}$
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 $\sqrt[n]{a}$ and $\sqrt[n]{b}$ are real numbers, $b not equal to 0,$ and for any integer $n greater than or equal to 2$ then,
  $\sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}}$ and $\frac{\sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[n]{\frac{a}{b}}$
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.

Practice Makes Perfect

Use the Product Property to Simplify Radical Expressions

In the following exercises, use the Product Property to simplify radical expressions.

$the square root of 27$

$3 the square root of 3$

$the square root of 80$

$the square root of 125$

$5 the square root of 5$

$the square root of 96$

$the square root of 147$

$7 the square root of 3$

$the square root of 450$

$the square root of 800$

$20 the square root of 2$

$the square root of 675$

ⓐ $the 4th root of 32$ ⓑ $the 5th root of 64$

ⓐ $2 the 4th root of 2$ ⓑ $2 the 5th root of 2$

ⓐ $the cube root of 625$ ⓑ $the 6th root of 128$

ⓐ $the 4th root of 64$ ⓑ $the cube root of 256$

ⓐ $2 the 4th root of 4$ ⓑ $4 the cube root of 4$

ⓐ $the 4th root of 3125$ ⓑ $the cube root of 81$

In the following exercises, simplify using absolute value signs as needed.

ⓐ $\sqrt{y^{11}}$ ⓑ $\sqrt[3]{r^{5}}$ ⓒ $\sqrt[4]{s^{10}}$

ⓐ $| y^{5} | \sqrt[]{y}$ ⓑ $r \sqrt[3]{r^{2}}$ ⓒ $s squared the 4th root of s squared$

ⓐ $\sqrt{m^{13}}$ ⓑ $\sqrt[5]{u^{7}}$ ⓒ $\sqrt[6]{v^{11}}$

ⓐ $\sqrt{n^{21}}$ ⓑ $\sqrt[3]{q^{8}}$ ⓒ $\sqrt[8]{n^{10}}$

ⓐ $n^{10} \sqrt{n}$ ⓑ $q^{2} \sqrt[3]{q^{2}}$
ⓒ $vertical bar n vertical bar the 8th root of n squared$

ⓐ $\sqrt{r^{25}}$ ⓑ $\sqrt[5]{p^{8}}$ ⓒ $\sqrt[4]{m^{5}}$

ⓐ $\sqrt{125 r^{13}}$ ⓑ $\sqrt[3]{108 x^{5}}$ ⓒ $\sqrt[4]{48 y^{6}}$

ⓐ $5 r^{6} \sqrt[]{5 r}$ ⓑ $3 x the cube root of 4 x squared$
ⓒ $2 vertical bar y vertical bar the 4th root of 3 y squared$

ⓐ $\sqrt{80 s^{15}}$ ⓑ $\sqrt[5]{96 a^{7}}$ ⓒ $\sqrt[6]{128 b^{7}}$

ⓐ $\sqrt{242 m^{23}}$ ⓑ $\sqrt[4]{405 m^{10}}$ ⓒ $\sqrt[5]{160 n^{8}}$

ⓐ $11 | m^{11} | \sqrt[]{2 m}$ ⓑ $3 m squared the 4th root of 5 m squared$ ⓒ $2 n the 5th root of 5 n cubed$

ⓐ $\sqrt{175 n^{13}}$ ⓑ $\sqrt[5]{512 p^{5}}$ ⓒ $\sqrt[4]{324 q^{7}}$

ⓐ $\sqrt{147 m^{7} n^{11}}$ ⓑ $\sqrt[3]{48 x^{6} y^{7}}$ ⓒ $\sqrt[4]{32 x^{5} y^{4}}$

ⓐ $7 | m^{3} n^{5} | \sqrt[]{3 m n}$ ⓑ $2 x squared y squared the cube root of 6 y$ ⓒ $2 vertical bar x y vertical bar the 4th root of 2 x$

ⓐ $the square root of 96 r cubed s cubed$ ⓑ $\sqrt[3]{80 x^{7} y^{6}}$ ⓒ $\sqrt[4]{80 x^{8} y^{9}}$

ⓐ $\sqrt{192 q^{3} r^{7}}$ ⓑ $\sqrt[3]{54 m^{9} n^{10}}$ ⓒ $\sqrt[4]{81 a^{9} b^{8}}$

ⓐ $8 vertical bar q r cubed vertical bar the square root of 3 q r$ ⓑ $3 m cubed n cubed the cube root of 2 n$ ⓒ $3 a squared b squared the 4th root of a$

ⓐ $\sqrt{150 m^{9} n^{3}}$ ⓑ $\sqrt[3]{81 p^{7} q^{8}}$ ⓒ $\sqrt[4]{162 c^{11} d^{12}}$

ⓐ $the cube root of −864$ ⓑ $the 4th root of −256$

ⓐ $−6 the cube root of 4$ ⓑ not real

ⓐ $the 5th root of −486$ ⓑ $the 6th root of −64$

ⓐ $the 5th root of −32$ ⓑ $the 8th root of −1$

ⓐ $−2$ ⓑ not real

ⓐ $the cube root of −8$ ⓑ $the 4th root of −16$

ⓐ $5 + \sqrt{12}$ ⓑ $\frac{10 - \sqrt{24}}{2}$

ⓐ $5 + 2 \sqrt{3}$ ⓑ $5 - \sqrt{6}$

ⓐ $8 + \sqrt{96}$ ⓑ $\frac{8 - \sqrt{80}}{4}$

ⓐ $1 + \sqrt{45}$ ⓑ $\frac{3 + \sqrt{90}}{3}$

ⓐ $1 + 3 \sqrt{5}$ ⓑ $1 + \sqrt{10}$

ⓐ $3 + \sqrt{125}$ ⓑ $\frac{15 + \sqrt{75}}{5}$

Use the Quotient Property to Simplify Radical Expressions

In the following exercises, use the Quotient Property to simplify square roots.

$\sqrt{\frac{96 x^{7}}{121}}$

$\frac{4 | x^{3} | \sqrt{6 x}}{11}$

$\sqrt{\frac{108 y^{4}}{49}}$

$\sqrt{\frac{300 m^{5}}{64}}$

$\frac{5 m^{2} \sqrt{3 m}}{4}$

$\sqrt{\frac{125 n^{7}}{169}}$

$\sqrt{\frac{98 r^{5}}{100}}$

$\frac{7 r^{2} \sqrt{2 r}}{10}$

$\sqrt{\frac{180 s^{10}}{144}}$

$\sqrt{\frac{28 q^{6}}{225}}$

$\frac{2 | q^{3} | \sqrt{7}}{15}$

$\sqrt{\frac{150 r^{3}}{256}}$

ⓐ $\frac{2 | p^{3} | \sqrt{7 p}}{| q |}$ ⓑ $\frac{3 s^{2} \sqrt[3]{3 s^{2}}}{t}$
ⓒ $\frac{2 | p^{3} | \sqrt[4]{4 p^{3}}}{| q^{3} |}$

ⓐ $\sqrt{\frac{32 x^{5} y^{3}}{18 x^{3} y}}$ ⓑ $\sqrt[3]{\frac{5 x^{6} y^{9}}{40 x^{5} y^{3}}}$ ⓒ $\sqrt[4]{\frac{5 a^{8} b^{6}}{80 a^{3} b^{2}}}$

ⓐ $\sqrt{\frac{75 r^{6} s^{8}}{48 r s^{4}}}$ ⓑ $\sqrt[3]{\frac{24 x^{8} y^{4}}{81 x^{2} y}}$ ⓒ $\sqrt[4]{\frac{32 m^{9} n^{2}}{162 m n^{2}}}$

ⓐ $\sqrt{\frac{27 p^{2} q}{108 p^{4} q^{3}}}$ ⓑ $\sqrt[3]{\frac{16 c^{5} d^{7}}{250 c^{2} d^{2}}}$ ⓒ $\sqrt[6]{\frac{2 m^{9} n^{7}}{128 m^{3} n}}$

ⓐ $\sqrt{\frac{50 r^{5} s^{2}}{128 r^{2} s^{6}}}$ ⓑ $\sqrt[3]{\frac{24 m^{9} n^{7}}{375 m^{4} n^{}}}$ ⓒ $\sqrt[4]{\frac{81 m^{2} n^{8}}{256 m^{1} n^{2}}}$

ⓐ $\frac{\sqrt{45 p^{9}}}{\sqrt{5 q^{2}}}$ ⓑ $\frac{\sqrt[4]{64}}{\sqrt[4]{2}}$ ⓒ $\frac{\sqrt[5]{128 x^{8}}}{\sqrt[5]{2 x^{2}}}$

ⓐ $\frac{\sqrt{80 q^{5}}}{\sqrt{5 q}}$ ⓑ $\frac{\sqrt[3]{−625}}{\sqrt[3]{5}}$ ⓒ $\frac{\sqrt[4]{80 m^{7}}}{\sqrt[4]{5 m}}$

ⓐ $5 vertical bar m cubed vertical bar$ ⓑ $5 the cube root of 5$
ⓒ $3 vertical bar y vertical bar the 4th root of 3 y squared$

Writing Exercises

Explain why $\sqrt{x^{4}} = x^{2} .$ Then explain why $\sqrt{x^{16}} = x^{8} .$

Answers will vary.

Explain why $7 + \sqrt{9}$ is not equal to $\sqrt{7 + 9} .$

Explain how you know that $\sqrt[5]{x^{10}} = x^{2} .$

Answers will vary.

Explain why $the 4th root of −64$ is not a real number but $the cube root of −64$ is.

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

This table has 3 rows and 4 columns. The first row is a header row and it labels each column. The first column header is “I can…”, the second is “Confidently”, the third is “With some help”, and the fourth is “No, I don’t get it”. Under the first column are the phrases “use the product property to simplify radical expressions” and “use the quotient property to simplify radical expressions”. The other columns are left blank so that the learner may indicate their mastery level for each topic.

ⓑ After reviewing this checklist, what will you do to become confident for all objectives?

Adapted from Intermediate Algebra 2e by OpenStax (openstax.org), licensed under CC BY-NC-SA 4.0. Changes were made. License: CC-BY-NC-SA-4.0.