Difference between revisions of "1966 AHSME Problems/Problem 12"
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== Problem == | == Problem == | ||
− | The number of real values of <math>x</math> | + | The number of real values of <math>x</math> that satisfy the equation <cmath>(2^{6x+3})(4^{3x+6})=8^{4x+5}</cmath> is: |
<math>\text{(A) zero} \qquad \text{(B) one} \qquad \text{(C) two} \qquad \text{(D) three} \qquad \text{(E) greater than 3}</math> | <math>\text{(A) zero} \qquad \text{(B) one} \qquad \text{(C) two} \qquad \text{(D) three} \qquad \text{(E) greater than 3}</math> | ||
== Solution == | == Solution == | ||
− | + | We know that | |
+ | <math>2^{6x+3}\cdot4^{3x+6}=2^{6x+3}\cdot(2^2)^{3x+6}=2^{6x+3}\cdot2^{6x+12}=2^{12x+15}</math>. | ||
+ | We also know that | ||
+ | <math>8^{4x+5}=(2^3)^{4x+5}=2^{12x+15}</math>. | ||
+ | There are infinite solutions to the equation <math>2^{12x+15}=2^{12x+15}</math>, so the answer is <math>\boxed{\text{(E) greater than 3}}</math>. | ||
== See also == | == See also == | ||
{{AHSME box|year=1966|num-b=10|num-a=12}} | {{AHSME box|year=1966|num-b=10|num-a=12}} | ||
− | [[Category:]] | + | [[Category:Introductory Algebra Problems]] |
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 19:48, 24 March 2023
Problem
The number of real values of that satisfy the equation
is:
Solution
We know that
.
We also know that
.
There are infinite solutions to the equation
, so the answer is
.
See also
1966 AHSME (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 • 26 • 27 • 28 • 29 • 30 | ||
All AHSME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.