# 11.2: Radical Expressions

**At Grade**Created by: CK-12

## Learning Objectives

At the end of this lesson, students will be able to:

- Use the product and quotient properties of radicals.
- Rationalize the denominator.
- Add and subtract radical expressions.
- Multiply radical expressions.
- Solve real-world problems using square root functions.

## Vocabulary

Terms introduced in this lesson:

- radical sign
- even roots, odd roots
- simplest radical form
- rationalizing the denominator

## Teaching Strategies and Tips

In this lesson, students learn that:

- Radicals reverse the operation of exponentiation.
- The index determines the kind of root.
- The square root is the only index which is not explicitly written but understood.

Use Example 1 to point out that even and odd indices handle negatives differently.

Additional Examples:

a. \begin{align*}\sqrt{-64}\end{align*} is not a real number.

b. \begin{align*}\sqrt[3]{-64}\end{align*} is defined and evaluates as \begin{align*}-4\end{align*}.

Use Example 2 to motivate rational exponents.

- One way to justify that \begin{align*}\sqrt[n]{a}=a^{1/n}\end{align*} is to use remind students of the power rule: \begin{align*}\Big(x^m\Big)^n=x^{mn}\end{align*}.
- Let \begin{align*}m=\frac{1}{2}\end{align*} and \begin{align*}n=2\end{align*}: \begin{align*}\Big(x^{\frac{1}{2}}\Big)^2=x^1=x\end{align*}.
- Therefore, \begin{align*}x^{1/2}\end{align*} is a number that when squared equals \begin{align*}x\end{align*}. Therefore, \begin{align*}x^{1/2}=\sqrt{x}\end{align*}.
- A similar argument holds in general for any index: \begin{align*}\sqrt[n]{a}=a^{1/n}\end{align*}.
- Using the power rule again: \begin{align*}a^{m/n}=a^{m\cdot \frac{1}{n}}=\Big(a^m\Big)^{\frac{1}{n}}=\sqrt[n]{a^m}\end{align*}.
- Therefore, \begin{align*}\sqrt[n]{a^n}=a^{n/n}\end{align*}.

Have students state the radical properties in words. This can help students learn the rules:

- The product rule for radicals: The square root of the product is the product of the square roots.
- The quotient rule for radicals: The square root of the quotient is the quotient of the square roots.

Rationalizing the denominator:

- Remind students to multiply the numerator
*and*denominator by the radical expression. “What you do to the top you do to the bottom.” - Point out that rationalizing the denominator is essentially multiplying by \begin{align*}1\end{align*}; therefore, the value of the original rational expression does not change.
- Have students seek a radical expression that when multiplied with the denominator results in a perfect power.
- In the case when the denominator contains two terms, one being a radical, a good choice for the rationalization is an expression whose product is a difference of squares.
- See
*Review Questions*26-33.

Encourage students to leave their answers in radical form unless otherwise specified.

- Radical form is an exact answer.
- If a decimal is needed, the final radical can be rounded.

Have students simplify *all* radicals to simplest form to ensure that all possible like terms in the expression are combined.

For students having a difficult time adding and simplifying radical expressions, draw the analogy with combining like terms in variable expressions. For example, the expressions \begin{align*}-2x +7x\end{align*} and \begin{align*}-2\sqrt{5}+7\sqrt{5}\end{align*} are essentially the same.

Teachers are encouraged to be specific about when a radical is in *simplified form*:

- No fractions occur in the radicand.

Example: The expression \begin{align*}x^2y\sqrt{\frac{x}{4}}\end{align*} can be simplified to \begin{align*}\frac{x^{\frac{5}{2}}y}{2}\end{align*}.

- No radicals are present in the denominator of a fraction.

Example: The expression \begin{align*}\frac{1}{\sqrt{x}}\end{align*} can be simplified to \begin{align*}\frac{\sqrt{x}}{x}\end{align*} or \begin{align*}x^{-\frac{1}{2}}\end{align*}. Decide whether to include negative exponents in simplified form.

- The index of a radical and the exponents on any expressions in the radicand do not have common factors.

Example: The expression \begin{align*}\sqrt[6]{x^2}\end{align*} can be simplified to \begin{align*}\sqrt[3]{x}\end{align*}.

- The exponents on any expressions in the radicand are less than the index.

Example: The expression \begin{align*}\sqrt[5]{x^9}\end{align*} can be simplified to \begin{align*}x\sqrt[5]{x^4}\end{align*}.

- The resulting expression has as few radicals as possible.

Example: The expression \begin{align*}\sqrt{5}+\sqrt{20}\end{align*} can be simplified to \begin{align*}3\sqrt{5}\end{align*}.

## Error Troubleshooting

In *Review Questions* 9-16 have students look for the highest possible perfect squares, cubes, fourth powers, etc. as indicated by the index of the radical. Suggest that they use factor trees as guides.

In *Review Questions* 14-16, have students treat the constants and variables separately.

General Tip: Remind students that when adding and subtracting radical expressions to combine only *like* radical terms (the same expression under the radical sign). This is analogous to combining *like terms* in variable expressions.

In *Review Question* 25, remind students to multiply the numbers outside the radical sign and the numbers inside the radical sign separately. Use the rule: \begin{align*}a\sqrt{b} \cdot c\sqrt{d}=ac\sqrt{bd}\end{align*}.