Radicals as Fractional Exponents
Algebra 1 - Unit 3, Day 3
Today's Objective
Do Now: Exponents & Roots
Simplify each expression.
\( (x^3)^2 = ? \)
\( \sqrt{49} = ? \)
\( \sqrt[3]{8} = ? \)
The Big Idea: Power over Root
A rational (fraction) exponent is just another way to write a radical.
\( b^{\frac{m}{n}} = (\sqrt[n]{b})^m \)
Remember: The numerator (m) is the Power. The denominator (n) is the Root.
Practice: Exponent to Radical Form
I Do
\( x^{\frac{2}{3}} \)
\( (\sqrt[3]{x})^2 \)
We Do
\( 5^{\frac{1}{2}} \)
\( \sqrt{5} \)
You Do
\( a^{\frac{4}{5}} \)
\( (\sqrt[5]{a})^4 \)
I Do: \( x^{\frac{2}{3}} \) = \( (\sqrt[3]{x})^2 \)
We Do: \( 5^{\frac{1}{2}} \) =
You Do: \( a^{\frac{4}{5}} \) =
Practice: Radical to Exponent Form
I Do
\( (\sqrt[4]{y})^3 \)
\( y^{\frac{3}{4}} \)
We Do
\( \sqrt{6} \)
\( 6^{\frac{1}{2}} \)
You Do
\( (\sqrt[5]{z})^2 \)
\( z^{\frac{2}{5}} \)
I Do: \( (\sqrt[4]{y})^3 \) = \( y^{\frac{3}{4}} \)
We Do: \( \sqrt{6} \) =
You Do: \( (\sqrt[5]{z})^2 \) =
Practice: Evaluating Expressions
I Do
\( 8^{\frac{2}{3}} \)
\( (\sqrt[3]{8})^2 \)
\( (2)^2 \)
\( 4 \)
We Do
\( 25^{\frac{3}{2}} \)
\( (\sqrt{25})^3 \)
\( (5)^3 \)
\( 125 \)
You Do
\( 16^{\frac{1}{4}} \)
\( \sqrt[4]{16} \)
\( 2 \)
I Do: \( 8^{\frac{2}{3}} \) = 4
We Do: \( 25^{\frac{3}{2}} \) =
You Do: \( 16^{\frac{1}{4}} \) =
Independent Practice
Complete the following problems. Start with the green section!
Rewrite or evaluate:
1. Rewrite \( x^{\frac{1}{4}} \) in radical form.
2. Evaluate \( 9^{\frac{1}{2}} \)
Rewrite or evaluate:
3. Rewrite \( (\sqrt[5]{y})^2 \) in exponent form.
4. Evaluate \( 27^{\frac{2}{3}} \)
Rewrite or evaluate:
5. Evaluate \( 32^{\frac{3}{5}} \)
Independent Practice
1. Rewrite \( x^{\frac{1}{4}} \) in radical form:
2. Evaluate \( 9^{\frac{1}{2}} \):
3. Rewrite \( (\sqrt[5]{y})^2 \) in exponent form:
4. Evaluate \( 27^{\frac{2}{3}} \):
5. Evaluate \( 32^{\frac{3}{5}} \):
Exit Ticket
Evaluate the following expression completely:
\( 64^{\frac{2}{3}} \)
\( (\sqrt[3]{64})^2 \)
\( (4)^2 \)
Answer: \( 16 \)