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## 8.2.1 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 real numbers $a$ and $m$, and $n≥2$,

$\sqrt[n]{a}$ is considered simplified if $a$ has no factors of $m^{n}$

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 $4$.

Similarly, $\sqrt[3]{4}$ is simplified because there are no perfect cube factors in $4$. But $\sqrt[3]{24}$ 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 $(ab)^{n}=a^{n}b^{n}$. The corresponding of Product Property of Roots says that $\sqrt[n]{ab}=\sqrt[n]{a} \cdot \sqrt[n]{b}$.

### PRODUCT PROPERTY OF NTH ROOTS

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

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

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

#### Example 1

Simplify: $\sqrt{98}$.

Solution
 Step 1. Find the largest factor in the radicand that is a perfect power of the inner.Rewrite the radicand as a product of two factors, using that factor. We see that $49$ is the largest factor of $98$ that has a power of $2$. In other words, $49$ is the largest perfect square factor of $98$.$98=49 \cdot 2$Always write the perfect square factor first. $\sqrt{98}$$\sqrt{49 \cdot 2} Step 2. Use the product rule to rewrite the radical as the product of two radicals. \sqrt{49} \cdot \sqrt{2} Step 3. Simplify the root of the perfect power. 7\sqrt{2} 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. The expression 7\sqrt{2} is very different from \sqrt[7]{2}. ### 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. 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: • \sqrt{500} • \sqrt[3]{16} • \sqrt[4]{243} Solution Part 1  \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} Part 2  \sqrt[3]{16} Rewrite the radicand as a product using the greatest perfect cube factor. 2^{3}=8 \sqrt[3]{8 \cdot 2} Rewrite the radical as the product of two radicals. \sqrt[3]{8} \cdot \sqrt[3]{2} Simplify. 2\sqrt[3]{2} Part 3  \sqrt[4]{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\sqrt[4]{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}} Solution Part 1  \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} Part 2  \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} Part 3  \sqrt[4]{x^{7}} Rewrite the radicand as a product using the largest 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. |x|\sqrt[4]{x^{3}} 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{72n^{7}} • \sqrt[3]{24x^{7}} • \sqrt[4]{80y^{14}} Solution Part 1  \sqrt{72n^{7}} Rewrite the radicand as a product using the largest perfect square factor. \sqrt{36n^{6} \cdot 2n} Rewrite the radical as the product of two radicals. \sqrt{36n^{6}} \cdot \sqrt{2n} Simplify. 6|n^{3}|\sqrt{2n} Part 2  \sqrt[3]{24x^{7}} Rewrite the radicand as a product using perfect cube factors. \sqrt[3]{8x^{6} \cdot 3x} Rewrite the radical as the product of two radicals. \sqrt[3]{8x^{6}} \cdot \sqrt[3]{3x} Simplify. 2x^{2}\sqrt[3]{3x} Part 3  \sqrt[4]{80y^{14}} Rewrite the radicand as a product using perfect fourth power factors. \sqrt[4]{16y^{12} \cdot 5y^{2}} Rewrite the radical as the product of two radicals. \sqrt[4]{16y^{12}} \cdot \sqrt[4]{5y^{2}} Rewrite the first radicand as (2y^{3})^{4}. \sqrt[4]{(2y^{3})^{4}} \cdot \sqrt[4]{5y^{2}} Simplify. 2|y^{3}|\sqrt[4]{5y^{2}} In the next example, we continue to use the same methods even though there are more than one variable under the radical. #### Example 5 • \sqrt{63u^{3}v^{5}} • \sqrt[3]{40x^{4}y^{5}} • \sqrt[4]{48x^{4}y^{7}} Solution Part 1  \sqrt{63u^{3}v^{5}} Rewrite the radicand as a product using the largest perfect square factor. \sqrt{9u^{2}v^{4} \cdot 7uv} Rewrite the radical as the product of two radicals. \sqrt{9u^{2}v^{4}} \cdot \sqrt{7uv} Rewrite the first radicand as (3uv^{2})^{2}. \sqrt{(3uv^{2})^{2}} \cdot \sqrt{7uv} Simplify. 3|u|v^{2}\sqrt{7uv} Part 2  \sqrt[3]{40x^{4}y^{5}} Rewrite the radicand as a product using the largest perfect cube factor. \sqrt[3]{8x^{3}y^{3} \cdot 5xy^{2}} Rewrite the radical as the product of two radicals. \sqrt[3]{8x^{3}y^{3}} \cdot \sqrt[3]{5xy^{2}} Rewrite the first radicand as (2xy)^{3}. \sqrt[3]{(2xy)^{3}} \cdot \sqrt[3]{5xy^{2}} Simplify. 2xy \sqrt[3]{5xy^{2}} Part 3  \sqrt[4]{48x^{4}y^{7}} Rewrite the radicand as a product using the largest perfect fourth power factor. \sqrt[4]{16x^{4}y^{4} \cdot 3y^{3}} Rewrite the radical as the product of two radicals. \sqrt[4]{16x^{4}y^{4}} \cdot \sqrt[4]{3y^{3}} Rewrite the first radicand as (2xy)^{4}. \sqrt[4]{(2xy)^{4}} \cdot \sqrt[4]{3y^{3}} Simplify. 2|xy| \sqrt[4]{3y^{3}} #### Example 6 • \sqrt[3]{-27} • \sqrt[4]{-16} Solution Part 1  \sqrt[3]{-27} Rewrite the radicand as a product using the perfect cube factors. \sqrt[3]{(-3)^{3}} Take the cube root. -3 Part 2  \sqrt[4]{-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 • 3+\sqrt{32} • \frac{4-\sqrt{48}}{2} Solution Part 1  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! Part 2  \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{\cancel{2}\cdot 2(1-\sqrt{3})}{\cancel{2}} Simplify. 2(1-\sqrt{3}) ## 8.2.2 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 • \sqrt{\frac{45}{80}} • \sqrt[3]{\frac{16}{54}} • \sqrt[4]{\frac{5}{80}} Solution Part 1  \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} Part 2  \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} Part 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, \boldsymbol{\frac{a^{m}}{a^{n}}=a^{m-n}, a≠0} #### Example 8 • \sqrt{\frac{m^{6}}{m^{4}}} • \sqrt[3]{\frac{a^{8}}{a^{5}}} • \sqrt[4]{\frac{a^{10}}{a^{2}}} Solution Part 1  \sqrt{\frac{m^{6}}{m^{4}}} Simplify inside the radical first. Divide the like bases by subtracting the exponents. \sqrt{m^{6-4}}$$\sqrt{m^{2}}$ Simplify. $|m|$

Part 2

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

Part 3

 $\sqrt[4]{\frac{a^{10}}{a^{2}}}$ 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.

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

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.

### QUOTIENT PROPERTY OF RADICAL EXPRESSIONS

If $\sqrt[n]{a}$ and $\sqrt[n]{b}$ are real numbers, $b≠0$, and for any integer $n≥2$ then,

$\sqrt[n]{\frac{a}{b}} = \frac{ \sqrt[n]{a}}{\sqrt[n]{b}}$ and $\frac{ \sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[n]{\frac{a}{b}}$

#### Example 9

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

Solution
 Step 1. Simplify the fraction in the radicand, if possible. $\frac{27m^{3}}{196}$ cannot be simplified. $\sqrt{\frac{27m^{3}}{196}}$ Step 2. Use the Quotient Property to rewrite the radical as the quotient of two radicals. Rewrite $\sqrt{\frac{27m^{3}}{196}}$ as the quotient of $\sqrt{27m^{3}}$ and $\sqrt{196}$ $\frac{\sqrt{27m^{3}}}{\sqrt{196}}$ Step 3. Simplify the radicals in the numerator and denominator. $9m^{2}$ and $196$ are perfect squares. $\frac{\sqrt{9m^{2}} \cdot \sqrt{3m}}{\sqrt{196}}$$\frac{3m\sqrt{3m}}{14}$

### HOW TO: 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 10

• $\sqrt{\frac{45x^{5}}{y^{4}}}$
• $\sqrt[3]{\frac{24x^{7}}{y^{3}}}$
• $\sqrt[4]{\frac{48x^{10}}{y^{8}}}$
Solution

Part 1

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

Part 2

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

Part 3

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

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

#### Example 11

• $\sqrt{\frac{18p^{5}q^{7}}{32pq^{2}}}$
• $\sqrt[3]{\frac{16x^{5}y^{7}}{54x^{2}y^{2}}}$
• $\sqrt[4]{\frac{5a^{8}b^{6}}{80a^{3}b^{2}}}$
Solution

Part 1

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

Part 2

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

Part 3

 $\sqrt[4]{\frac{5a^{8}b^{6}}{80a^{3}b^{2}}}$ 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. $\frac{|ab|\sqrt[4]{a}}{2}$

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 12

Simplify:

• $\frac{\sqrt{48a^{7}}}{\sqrt{3a}}$
• $\frac{\sqrt[3]{-108}}{\sqrt[3]{2}}$
• $\frac{\sqrt[4]{96x^{7}}}{\sqrt[4]{3x^{2}}}$
Solution

Part 1

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

Part 2

 $\frac{\sqrt[3]{-108}}{\sqrt[3]{2}}$ The denominator cannot be simplified, so use the Quotient Property to write as one radical. $\sqrt[3]{\frac{-108}{2}}$ Simplify the fraction under one radical. $\sqrt[3]{-54}$ 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. $-3\sqrt[3]{2}$

Part 3

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