Difference between revisions of "2022 AMC 10A Problems/Problem 11"
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<math>m = 3</math> and <math>m = 4</math> | <math>m = 3</math> and <math>m = 4</math> | ||
− | Since our two values for m are <math>3</math> and <math>4</math>, our final answer is <math>3+4 = \boxed{\textbf{(C)}7}</math> | + | Since our two values for m are <math>3</math> and <math>4</math>, our final answer is <math>3+4 = \boxed{\textbf{(C) } 7}</math> |
− | - abed_nadir | + | - abed_nadir |
==Video Solution 1 == | ==Video Solution 1 == |
Revision as of 04:47, 14 October 2023
Contents
[hide]Problem
Ted mistakenly wrote as What is the sum of all real numbers for which these two expressions have the same value?
Solution 1
We are given that Converting everything into powers of we have We multiply both sides by , then rearrange as By Vieta's Formulas, the sum of such values of is
Note that or from the quadratic equation above.
~MRENTHUSIASM
~KingRavi
Solution 2
Since surd roots are conventionally positive integers, assume is an integer, so can only be , , , , , and . . Testing out , we see that only and work. Hence, .
~MrThinker
Solution 3 (Logarithms)
We can rewrite the equation using fractional exponents and take logarithms of both sides:
We can then use the additive properties of logarithms to split them up:
Using the power rule, the fact that , and bringing the exponents down, we get:
and
Since our two values for m are and , our final answer is
- abed_nadir
Video Solution 1
~Education, the Study of Everything
Video Solution (Easy)
~Whiz
See Also
2022 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 10 |
Followed by Problem 12 | |
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 |
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