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11.2: Radical Expressions

Difficulty Level: At Grade Created by: CK-12
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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.


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*}.

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