Positive Rational Exponents: IM Alg2.3.4
[url=https://im.kendallhunt.com/HS/teachers/3/3/4/preparation.html]“Positive 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
- Alg2.3.4 Lesson
- IM Alg2.3.4 Lesson: Positive Rational Exponents
- Alg2.3.4 Practice
- IM Alg2.3.4 Practice: Positive Rational Exponents
IM Alg2.3.4 Lesson: Positive Rational Exponents
Find the value of each expression mentally.
[math]\frac{1}{2}\cdot5\cdot4[/math]
[math]\frac{5}{2}\cdot4[/math]
[math]\frac{2}{3}\cdot7\cdot\frac{3}{2}[/math]
[math]7\cdot\frac{5}{3}\cdot\frac{3}{7}[/math]
Use exponent rules to explain why these expressions are equal to each other:
[math](5^{\frac{1}{3}})^2[/math] [math](5^2)^{\frac{1}{3}}[/math]
Write [math]5^{\frac{2}{3}}[/math] using radicals.[br]
Write [math]5^{\frac{4}{3}}[/math] using radicals. Show your reasoning using exponent rules.[br]
Complete the table as much as you can without using a calculator. (You should be able to fill in three spaces.)
Plot the points that you filled in. Connect the points as smoothly as you can.
Use this 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.
[size=150]Let’s investigate [math]2^{\frac{1}{3}}[/math]:[/size][br][br]Write [math]2^{\frac{1}{3}}[/math] using radical notation.[br]
What is [math](2^{\frac{1}{3}})^3[/math]?[br]
Raise your estimate from the table of [math]2^{\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^{\frac{2}{3}}[/math]:[/size][br][br]Write [math]2^{\frac{2}{3}}[/math] using radical notation.[br]
What is the value of [math](2^{\frac{2}{3}})^3[/math]?[br]
Raise your estimate from the table of [math]2^{\frac{2}{3}}[/math] to the third power. What should it be? How close did you get?[br]
[size=150]Answer these questions using the fact that [math](1.26)^3=2.000376[/math][/size][br][br]Explain why [math]\sqrt[3]{2}[/math] is very close to [math]1.26[/math]. Is it larger or smaller than [math]1.26[/math]?[br]
Is it possible to write [math]\sqrt[3]{2}[/math] exactly with a finite decimal expansion? Explain how you know.[br]
IM Alg2.3.4 Practice: Positive Rational Exponents
[size=150]Evaluate [math]8^{\frac{5}{3}}[/math].[/size]
[size=150]Select [b]all[/b] expressions that are equal to [math]64^{\frac{3}{2}}[/math][/size]
[size=150]Write the expression [math]17^{\frac{4}{3}}[/math] using radicals.[/size]
[size=150]An arithmetic sequence [math]k[/math] starts 4, 13, . . . . Explain how you would calculate the value of the 5,000th term.[/size]
Select [b]all[/b] items equivalent to [math]\sqrt{24}[/math].
[size=150]Which expression is equivalent to [math]23^{\frac{1}{2}}[/math]?[/size]