Expressions with Exponents: IM 6.6.13
[url=https://im.kendallhunt.com/MS/teachers/1/6/13/preparation.html]“Expressions with Exponents”[/url] from IM Grade 6 by [url=https://www.geogebra.org/m/gus8ffvr]Open Up Resources[/url] and Illustrative Mathematics. Licensed under the [url=https://creativecommons.org/licenses/by/4.0/]Creative Commons Attribution 4.0 license[/url].
Table of Contents
- Lesson 6.6.13
- IM 6.6.13 Lesson: Expressions with Exponents
- Practice 6.6.13
- IM 6.6.13 Practice: Expressions with Exponents
IM 6.6.13 Lesson: Expressions with Exponents
Which one doesn’t belong?
[table][tr][td][math]2\cdot2\cdot2\cdot2[/math][/td][td][math]2^4[/math][/td][/tr][tr][td][math]16[/math][/td][td][math]4\cdot2[/math][/td][/tr][/table]
Decide whether each equation is true or false, and explain how you know.
[math]2^4=2\cdot4[/math]
Explain how you know.
[math]3+3+3+3+3=3^5[/math]
Explain how you know.
[math]5^3=5\cdot5\cdot5[/math]
Explain how you know.
[math]2^3=3^2[/math]
Explain how you know.
[math]16^1=8^2[/math]
Explain how you know.
[math]\frac{1}{2}\cdot\frac{1}{2}\cdot\frac{1}{2}\cdot\frac{1}{2}=4\cdot\frac{1}{2}[/math]
Explain how you know.
[math]\left(\frac{1}{2}\right)^4=\frac{1}{8}[/math]
Explain how you know.
[math]8^2=4^3[/math]
Explain how you know.
[size=150]In the list, find expressions that are equivalent to each other and explain to your partner why they are equivalent. Your partner listens to your explanation. If you disagree, explain your reasoning until you agree. Switch roles for each list. (There may be more than two equivalent expressions in each list.)[/size][br][br][list][*][math]5\cdot5[/math][/*][*][math]2^5[/math][/*][*][math]5^2[/math][/*][*][math]2\cdot5[/math][/*][/list]
[list][*][math]4^3[/math][br][/*][*][math]3^4[/math][br][/*][*][math]4\cdot4\cdot4[/math][br][/*][*][math]4+4+4[/math] [/*][/list]
[list][*][math]6+6+6[/math] [/*][*][math]6^3[/math] [/*][*][math]3^6[/math] [/*][*][math]3\cdot6[/math][/*][/list]
[list][*][math]11^5[/math][br][/*][*][math]11\cdot11\cdot11\cdot11\cdot11[/math][br][/*][*][math]11\cdot5[/math][br][/*][*][math]5^{11}[/math][/*][/list]
[list][*][math]\frac{1}{5}\cdot\frac{1}{5}\cdot\frac{1}{5}[/math][br][/*][*][math]\left(\frac{1}{5}\right)^3[/math][br][/*][*][math]\frac{1}{15}[/math][br][/*][*][math]\frac{1}{125}[/math] [/*][/list]
[list][*][math]\left(\frac{5}{3}\right)^2[/math][br][/*][*][math]\left(\frac{3}{5}\right)^2[/math][br][/*][*][math]\frac{10}{6}[/math][br][/*][*][math]\frac{25}{9}[/math][/*][/list]
What is the last digit of [math]3^{1,000}[/math]? Show or explain your reasoning.
IM 6.6.13 Practice: Expressions with Exponents
Select [b]all [/b]expressions that are equal to [math]3\cdot3\cdot3\cdot3\cdot3[/math].
Noah starts with 0 and then adds the number 5 four times. Diego starts with 1 and then multiplies by the number 5 four times. Decide whether the expression it is equal to Noah’s result, Diego’s result, or neither.
[math]4\cdot5[/math]
[math]4+5[/math]
[math]4^5[/math]
[math]5^4[/math]
Decide whether the equation is true or false.[br][br][math]9\cdot9\cdot3=3^5[/math]
Explain how you know.
Decide whether each equation is true or false, and explain how you know.
[math]7+7+7=3+3+3+3+3+3+3[/math]
Explain how you know.
[math]\frac{1}{7}\cdot\frac{1}{7}\cdot\frac{1}{7}=\frac{3}{7}[/math]
Explain how you know.
[math]4^1=4\cdot1[/math]
Explain how you know.
[math]6+6+6=6^3[/math]
Explain how you know.
What is the area of a square with side lengths of [math]\frac{3}{5}[/math] units?
What is the side length of a square with area [math]\frac{1}{16}[/math] square units?
What is the volume of a cube with edge lengths of [math]\frac{2}{3}[/math] units?
What is the edge length of a cube with volume [math]\frac{27}{64}[/math] cubic units?
Select [b]all [/b]the expressions that represent the area of the shaded rectangle.[br][img]data:image/png;base64,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[/img]
A ticket at a movie theater costs $8.50. One night, the theater had $29,886 in ticket sales.
Estimate about how many tickets the theater sold. Explain your reasoning.[br]
How many tickets did the theater sell? Explain your reasoning.
A fence is being built around a rectangular garden that is [math]8\frac{1}{2}[/math] feet by [math]6\frac{1}{3}[/math] feet. Fencing comes in panels. Each panel is [math]\frac{2}{3}[/math] of a foot wide. How many panels are needed? Explain or show your reasoning.