### Raising a Product or a Quotient to a Power

A radical reverses the operation of raising a number to a power. For example, the square of 4 is \begin{align*}4^2 = 4 \cdot 4 = 16\end{align*}, and so the square root of 16 is 4. The symbol for a square root is \begin{align*}\sqrt{\;\;}\end{align*}. This symbol is also called the **radical sign.**

In addition to square roots, we can also take cube roots, fourth roots, and so on. For example, since 64 is the cube of 4, 4 is the cube root of 64.

\begin{align*}\sqrt[3]{64} = 4 \qquad \text{since} \qquad 4^3 = 4 \cdot 4 \cdot 4 = 64\end{align*}

We put an index number in the top left corner of the radical sign to show which root of the number we are seeking. Square roots have an index of 2, but we usually don’t bother to write that out.

\begin{align*}\sqrt[2]{36} = \sqrt{36} = 6\end{align*}

The cube root of a number gives a number which when raised to the power three gives the number under the radical sign. The fourth root of number gives a number which when raised to the power four gives the number under the radical sign:

\begin{align*}\sqrt[4]{81} = 3 \qquad \text{since} \qquad 3^4 = 3 \cdot 3 \cdot 3 \cdot 3 = 81\end{align*}

And so on for any power we can name.

**Even and Odd Roots**

Radical expressions that have even indices are called **even roots** and radical expressions that have odd indices are called **odd roots**. There is a very important difference between even and odd roots, because they give drastically different results when the number inside the radical sign is negative.

Any real number raised to an even power results in a positive answer. Therefore, when the index of a radical is even, the number inside the radical sign must be non-negative in order to get a real answer.

On the other hand, a positive number raised to an odd power is positive and a negative number raised to an odd power is negative. Thus, a negative number inside the radical sign is not a problem. It just results in a negative answer.

#### Evaluating Radical Expressions

Evaluate each radical expression.

a) \begin{align*}\sqrt{121}\end{align*}

\begin{align*}\sqrt{121} = 11\end{align*}

b) \begin{align*}\sqrt[3]{125}\end{align*}

\begin{align*}\sqrt[3]{125} = 5\end{align*}

c) \begin{align*}\sqrt[4]{-625}\end{align*}

\begin{align*}\sqrt[4]{-625}\end{align*} is not a real number

d) \begin{align*}\sqrt[5]{-32}\end{align*}

\begin{align*}\sqrt[5]{-32} = -2\end{align*}

**Using the Product and Quotient Properties of Radicals**

Radicals can be re-written as rational powers. The radical: \begin{align*}\sqrt[m]{a^n}\end{align*} is defined as \begin{align*}a^{\frac{n}{m}}\end{align*}.

Write each expression as an exponent with a rational value for the exponent.

a) \begin{align*}\sqrt{5}\end{align*}

\begin{align*}\sqrt{5} = 5^{\frac{1}{2}}\end{align*}

b) \begin{align*}\sqrt[4]{a}\end{align*}

\begin{align*}\sqrt[4]{a} = a^{\frac{1}{4}}\end{align*}

c) \begin{align*} \sqrt[3]{4xy}\end{align*}

\begin{align*} \sqrt[3]{4xy} = (4xy)^{\frac{1}{3}}\end{align*}

d) \begin{align*}\sqrt[6]{x^5}\end{align*}

\begin{align*}\sqrt[6]{x^5} = x^{\frac{5}{6}}\end{align*}

As a result of this property, for any non-negative number \begin{align*}a\end{align*} we know that \begin{align*}\sqrt[n]{a^n} = a^{\frac{n}{n}} = a\end{align*}.

Since roots of numbers can be treated as powers, we can use exponent rules to simplify and evaluate radical expressions. Let’s review the product and quotient rule of exponents.

\begin{align*}\text{Raising a product to a power:} && (x \cdot y)^n & = x^n \cdot y^n\\ \text{Raising a quotient to a power:} && \left(\frac{x}{y} \right)^n & = \frac{x^n}{y^n}\end{align*}

In radical notation, these properties are written as

\begin{align*}\text{Raising a product to a power:} && \sqrt[m]{x \cdot y} & = \sqrt[m]{x} \ \cdot \sqrt[m]{y}\\ \text{Raising a quotient to a power:} && \sqrt[m]{\frac{x}{y}} & = \frac{\sqrt[m]{x}}{\sqrt[m]{y}}\end{align*}

A very important application of these rules is reducing a radical expression to its simplest form. This means that we apply the root on all the factors of the number that are perfect roots and leave all factors that are not perfect roots inside the radical sign.

For example, in the expression \begin{align*}\sqrt{16}\end{align*}, the number 16 is a perfect square because \begin{align*}16 = 4^2\end{align*}. This means that we can simplify it as follows:

\begin{align*}\sqrt{16} = \sqrt{4^2} = 4\end{align*}

Thus, the square root disappears completely.

On the other hand, in the expression \begin{align*}\sqrt{32}\end{align*}, the number 32 is not a perfect square, so we can’t just remove the square root. However, we notice that \begin{align*}32 = 16 \cdot 2\end{align*}, so we can write 32 as the product of a perfect square and another number. Thus,

\begin{align*}\sqrt{32} = \sqrt{16 \cdot 2}\end{align*}

If we apply the “raising a product to a power” rule we get:

\begin{align*}\sqrt{32} = \sqrt{16 \cdot 2} = \sqrt{16} \ \cdot \sqrt{2}\end{align*}

Since \begin{align*}\sqrt{16} = 4\end{align*}, we get: \begin{align*}\sqrt{32} = 4 \cdot \sqrt{2} = \underline{\underline{4 \sqrt{2}}}\end{align*}

#### Writing Expressions in the Simplest Radical Form

1. Write the following expressions in the simplest radical form.

The strategy is to write the number under the square root as the product of a perfect square and another number. The goal is to find the highest perfect square possible; if we don’t find it right away, we just repeat the procedure until we can’t simplify any longer.

a) \begin{align*}\sqrt{8}\end{align*}

\begin{align*}\text{We can write} \ 8 = 4 \cdot 2, \ \text{so} \sqrt{8} = \sqrt{4 \cdot 2}.\\ \text{With the Raising a product to a power rule, that becomes} \sqrt{4} \ \cdot \sqrt{2}.\\ \text{Evaluate} \ \sqrt{4} \ \text{and we're left with } \underline{\underline{2\sqrt{2}}}.\end{align*}

b) \begin{align*} \sqrt{50}\end{align*}

\begin{align*}\text{We can write} \ 50 = 25 \cdot 2, \ \text{so:} \sqrt{50} = \sqrt{25 \cdot 2}\\ \text{Use the Raising a product to a power rule:} =\sqrt{25} \ \cdot \sqrt{2} = \underline{\underline{5 \sqrt{2}}}\end{align*}

c) \begin{align*}\sqrt{\frac{125}{72}}\end{align*}

\begin{align*}\text{Use the Raising a quotient to a power rule to separate the fraction:} \sqrt{\frac{125}{72}} = \frac{\sqrt{125}}{\sqrt{72}}\\ \text{Re-write each radical as a product of a perfect square and another number:} = \frac{ \sqrt{25 \cdot 5}}{\sqrt{36 \cdot 2}} = \frac{5 \sqrt{5}}{6 \sqrt{2}}\end{align*}

The same method can be applied to reduce radicals of different indices to their simplest form.

2. Write the following expression in the simplest radical form.

In these cases we look for the highest possible perfect cube, fourth power, etc. as indicated by the index of the radical.

a) \begin{align*}\sqrt[3]{40}\end{align*}

Here we are looking for the product of the highest perfect cube and another number. We write: \begin{align*}\sqrt[3]{40} = \sqrt[3]{8 \cdot 5} = \sqrt[3]{8} \ \cdot \sqrt[3]{5} = 2 \sqrt[3]{5}\end{align*}

b) \begin{align*}\sqrt[4]{\frac{162}{80}}\end{align*}

Here we are looking for the product of the highest perfect fourth power and another number.

\begin{align*}\text{Re-write as the quotient of two radicals:} && \sqrt[4]{\frac{162}{80}} & = \frac{\sqrt[4]{162}}{\sqrt[4]{80}}\\ \text{Simplify each radical separately:} && & = \frac{\sqrt[4]{81 \cdot 2}}{\sqrt[4]{16 \cdot 5}} = \frac{\sqrt[4]{81} \ \cdot \sqrt[4]{2}} {\sqrt[4]{16} \ \cdot \sqrt[4]{5}} = \frac{3 \sqrt[4]{2}}{2 \sqrt[4]{5}}\\ \text{Recombine the fraction under one radical sign:} && & = \frac{3}{2} \sqrt[4]{\frac{2}{5}}\end{align*}

c) \begin{align*}\sqrt[3]{135}\end{align*}

Here we are looking for the product of the highest perfect cube root and another number. Often it’s not very easy to identify the perfect root in the expression under the radical sign. In this case, we can factor the number under the radical sign completely by using a factor tree:

We see that \begin{align*}135 = 3 \cdot 3 \cdot 3 \cdot 5 = 3^3 \cdot 5\end{align*}. Therefore \begin{align*}\sqrt[3]{135} = \sqrt[3]{3^3 \cdot 5} = \sqrt[3]{3^3} \ \cdot \sqrt[3]{5} = 3 \sqrt[3]{5}\end{align*}.

Now let’s see some examples involving variables.

### Examples

Write the following expressions in the simplest radical form.

Treat constants and each variable separately and write each expression as the products of a perfect power as indicated by the index of the radical and another number.

#### Example 1

\begin{align*}\sqrt{12x^3 y^5}\end{align*}\begin{align*}\text{Re-write as a product of radicals:} \sqrt{12x^3y^5} = \sqrt{12} \ \cdot \sqrt{x^3} \ \cdot \sqrt{y^5}\\ \text{Simplify each radical separately:} \left(\sqrt{4 \cdot 3}\right ) \cdot \left( \sqrt{x^2 \cdot x}\right ) \cdot \left (\sqrt{y^4 \cdot y}\right ) = \left (2 \sqrt{3}\right ) \cdot \left (x \sqrt{x}\right ) \cdot \left (y^2 \sqrt{y}\right )\\ \text{Combine all terms outside and inside the radical sign:} =2xy^2 \sqrt{3xy}\end{align*}

#### Example 2

\begin{align*}\sqrt[4]{\frac{1250x^7}{405y^9}}\end{align*}\begin{align*}\text{Re-write as a quotient of radicals:} \sqrt[4]{\frac{1250x^7}{405y^9}} = \frac{\sqrt[4]{1250x^7}}{\sqrt[4]{405y^9}}\\ \text{Simplify each radical separately:} = \frac{\sqrt[4]{625 \cdot 2} \ \cdot \sqrt[4]{x^4 \cdot x^3}}{\sqrt[4]{81 \cdot 5} \ \cdot \sqrt[4]{y^4 \cdot y^4 \cdot y}} = \frac{5 \sqrt[4]{2} \cdot x \cdot \sqrt[4]{x^3}}{3 \sqrt[4]{5} \cdot y \cdot y \ \cdot \sqrt[4]{y}} = \frac{5x \sqrt[4]{2x^3}}{3y^2 \sqrt[4]{5y}}\\ \text{Recombine fraction under one radical sign:} = \frac{5x}{3y^2} \sqrt[4]{\frac{2x^3}{5y}}\end{align*}

### Review

Evaluate each radical expression.

- \begin{align*}\sqrt{169}\end{align*}
- \begin{align*}\sqrt[4]{-81}\end{align*}
- \begin{align*}\sqrt[3]{-125}\end{align*}
- \begin{align*}\sqrt[5]{1024}\end{align*}

Write each expression as a rational exponent.

- \begin{align*}\sqrt[3]{14}\end{align*}
- \begin{align*}\sqrt[4]{zw}\end{align*}
- \begin{align*}\sqrt{a}\end{align*}
- \begin{align*}\sqrt[9]{y^3}\end{align*}

Write the following expressions in simplest radical form.

- \begin{align*}\sqrt{24}\end{align*}
- \begin{align*}\sqrt{300}\end{align*}
- \begin{align*}\sqrt[5]{96}\end{align*}
- \begin{align*}\sqrt{\frac{240}{567}}\end{align*}
- \begin{align*}\sqrt[3]{500}\end{align*}
- \begin{align*}\sqrt[6]{64x^8}\end{align*}
- \begin{align*}\sqrt[3]{48a^3 b^7}\end{align*}
- \begin{align*}\sqrt[3]{\frac{16x^5}{135y^4}}\end{align*}

### Review (Answers)

To view the Review answers, open this PDF file and look for section 11.3.

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