Negative Rational Exponents: IM Alg2.3.5
[url=https://im.kendallhunt.com/HS/teachers/3/3/5/preparation.html]“Negative Rational Exponents”[/url] from IM Algebra II © 2019 [url=https://www.illustrativemathematics.org/]Illustrative Mathematics[/url]. Licensed under the [url=https://creativecommons.org/licenses/by/4.0/]Creative Commons Attribution 4.0[/url] license.
Table of Contents
- Alg2.3.5 Lesson
- IM Alg2.3.5 Lesson: Negative Rational Exponents
- Alg2.3.5 Practice
- IM Alg2.3.5 Practice: Negative Rational Exponents
IM Alg2.3.5 Lesson: Negative Rational Exponents
Evaluate mentally.
[math]9^2[/math]
[math]9^{-2}[/math][br]
[math]9^{\frac{1}{2}}[/math]
[math]9^{-\frac{1}{2}}[/math]
Complete the table as much as you can without using a calculator. (You should be able to fill in three spaces.)
Plot these powers of 2 in the coordinate plane. Connect the points as smoothly as you can.
Use your graph of [math]y=2^x[/math] to estimate the value of the other powers in the table, and write your estimates in the table.[br]
[size=150]Let’s investigate [math]2^{\text{-}\frac{1}{3}}[/math].[/size][br][br]Write [math]2^{\text{-}\frac{1}{3}}[/math] using radical notation.
What is the value of [math](2^{\text{-}\frac{1}{3}})^3[/math]?[br]
Raise your estimate of [math]2^{\text{-}\frac{1}{3}}[/math] to the third power. What should it be? [br]
How close did you get?
[size=150]Let’s investigate [math]2^{\text{-}\frac{2}{3}}[/math].[br][/size][br]Write [math]2^{\text{-}\frac{2}{3}}[/math] using radical notation.
What is [math](2^{\text{-}\frac{2}{3}})^3[/math]?[br]
Raise your estimate of [math]2^{-\frac{2}{3}}[/math] to the third power. What should it be? [br]
How close did you get?
For each set of 3 numbers, cross out the expression that is not equal to the other two expressions.
For each expression, write an equivalent expression using radicals.
[math]17^{\frac{3}{2}}[/math]
[math]31^{\frac{3}{2}}[/math]
For each expression, write an equivalent expression using only exponents.
[math](\sqrt{3})^4[/math]
[math]\frac{1}{(\sqrt[3]{5})^6}[/math]
Write two different expressions that involve only roots and powers of 2 which are equivalent to [math]\frac{4^{\frac{2}{3}}}{8^{\frac{1}{4}}}[/math].
Match expressions into groups according to whether they are equal. Be prepared to explain your reasoning.
IM Alg2.3.5 Practice: Negative Rational Exponents
[size=150]Write each expression in the form [math]a^b[/math], without using any radicals.[/size][br][math]\sqrt{5^9}[/math]
[math]\frac{1}{\sqrt[3]{12}}[/math]
Write [math]32^{\text{-}\frac{2}{5}}[/math] without using exponents or radicals.
Match the equivalent expressions.
Complete the table. Use powers of 27 in the top row and radicals or rational numbers in the bottom row.
What are the solutions to the equation [math](x-1)(x+2)=\text{-}2[/math]?
Use exponent rules to explain why [math]\left(\sqrt{5}\right)^3=\sqrt{5^3}[/math].