Difference between revisions of "2021 Fall AMC 12B Problems/Problem 4"
m |
Abed nadir (talk | contribs) m (→Solution 3) |
||
| (20 intermediate revisions by 11 users not shown) | |||
| Line 1: | Line 1: | ||
| − | {{duplicate|[[2021 Fall AMC 10B Problems | + | {{duplicate|[[2021 Fall AMC 10B Problems/Problem 5|2021 Fall AMC 10B #5]] and [[2021 Fall AMC 12B Problems/Problem 4|2021 Fall AMC 12B #4]]}} |
== Problem == | == Problem == | ||
Let <math>n=8^{2022}</math>. Which of the following is equal to <math>\frac{n}{4}?</math> | Let <math>n=8^{2022}</math>. Which of the following is equal to <math>\frac{n}{4}?</math> | ||
| − | <math> | + | <math>\textbf{(A)}\: 4^{1010}\qquad\textbf{(B)} \: 2^{2022}\qquad\textbf{(C)} \: 8^{2018}\qquad\textbf{(D)} \: 4^{3031}\qquad\textbf{(E)} \: 4^{3032}</math> |
==Solution 1== | ==Solution 1== | ||
| − | We have <cmath>n=8^{2022}= \left(8^\frac{2}{3}\right)^{ | + | We have <cmath>n=8^{2022}= \left(8^\frac{2}{3}\right)^{3033}=4^{3033}.</cmath> |
| − | Therefore, <cmath>\frac{n}4=\boxed{ | + | Therefore, <cmath>\frac{n}4=\boxed{\textbf{(E)} \: 4^{3032}}.</cmath> |
| − | |||
~kingofpineapplz | ~kingofpineapplz | ||
==Solution 2== | ==Solution 2== | ||
| − | The requested value is <cmath>\frac{8^{2022}}{4} = \frac{2^{6066}}{4} = \frac{2^{6066}}{2^2} = 2^{6064} = 4^{3032}.</cmath> | + | The requested value is <cmath>\frac{8^{2022}}{4} = \frac{2^{6066}}{4} = \frac{2^{6066}}{2^2} = 2^{6064} = \boxed{\textbf{(E)} \: 4^{3032}}.</cmath> |
| + | ~NH14 | ||
| + | |||
| + | ==Solution 3== | ||
| + | |||
| + | If we rewrite everything in powers of 2, we get: | ||
| + | <math>n = 8^{2022} = (2^{3})^{2022} = 2^{6064} = (4^{\frac{1}{2}})^{6064} = 4^{\frac{6064}{2}} = \boxed{\textbf{(E)} \: 4^{3032}}.</math> | ||
| + | |||
| + | - abed_nadir (youtube.com/@indianmathguy) | ||
| + | |||
| + | ==Video Solution by Interstigation== | ||
| + | https://youtu.be/p9_RH4s-kBA?t=429 | ||
| + | |||
| + | ==Video Solution (Just 3 min!)== | ||
| + | https://youtu.be/480KnrVnbOc | ||
| + | |||
| + | ~Education, the Study of Everything | ||
| + | |||
| + | ==Video Solution by WhyMath== | ||
| + | https://youtu.be/iXX4WtMKU_g | ||
| + | |||
| + | ~savannahsolver | ||
| + | ==Video Solution by TheBeautyofMath== | ||
| + | For AMC 10: https://youtu.be/lC7naDZ1Eu4?t=882 | ||
| − | ~ | + | For AMC 12: https://youtu.be/yaE5aAmeesc?t=590 |
| + | |||
| + | ~IceMatrix | ||
| + | |||
| + | ==See Also== | ||
| + | {{AMC10 box|year=2021 Fall|ab=B|num-a=6|num-b=4}} | ||
| + | {{AMC12 box|year=2021 Fall|ab=B|num-a=5|num-b=3}} | ||
| + | |||
| + | [[Category:Introductory Algebra Problems]] | ||
| + | {{MAA Notice}} | ||
Revision as of 03:27, 12 November 2023
- The following problem is from both the 2021 Fall AMC 10B #5 and 2021 Fall AMC 12B #4, so both problems redirect to this page.
Contents
Problem
Let 
. Which of the following is equal to 
Solution 1
We have ![\[n=8^{2022}= \left(8^\frac{2}{3}\right)^{3033}=4^{3033}.\]](http://latex.artofproblemsolving.com/f/f/8/ff8f511842bf3275a1d48df6d5f8adcd18102756.png)
Therefore, ![\[\frac{n}4=\boxed{\textbf{(E)} \: 4^{3032}}.\]](http://latex.artofproblemsolving.com/4/4/c/44cf34fd04107d751424dd4df3f40daae9b744e1.png)
~kingofpineapplz
Solution 2
The requested value is ![\[\frac{8^{2022}}{4} = \frac{2^{6066}}{4} = \frac{2^{6066}}{2^2} = 2^{6064} = \boxed{\textbf{(E)} \: 4^{3032}}.\]](http://latex.artofproblemsolving.com/7/d/8/7d81ce2dc12bcdcd5021c0d085e79c9f61a4ce52.png)
~NH14
Solution 3
If we rewrite everything in powers of 2, we get:
- abed_nadir (youtube.com/@indianmathguy)
Video Solution by Interstigation
https://youtu.be/p9_RH4s-kBA?t=429
Video Solution (Just 3 min!)
~Education, the Study of Everything
Video Solution by WhyMath
~savannahsolver
Video Solution by TheBeautyofMath
For AMC 10: https://youtu.be/lC7naDZ1Eu4?t=882
For AMC 12: https://youtu.be/yaE5aAmeesc?t=590
~IceMatrix
See Also
| 2021 Fall AMC 10B (Problems • Answer Key • Resources) | ||
| Preceded by Problem 4 |
Followed by Problem 6 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
| All AMC 10 Problems and Solutions | ||
| 2021 Fall AMC 12B (Problems • Answer Key • Resources) | |
| Preceded by Problem 3 |
Followed by Problem 5 |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
| All AMC 12 Problems and Solutions | |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
