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Difference between revisions of "1966 AHSME Problems/Problem 12"

(Created page with "== Problem == The number of real values of <math>x</math> which satisfy the equation <cmath>(2^{6x+3})(4^{3x+6})=8^{4x+5}</cmath> is: <math>\text{(A) zero} \qquad \text{(B) on...")
 
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== Problem ==
 
== Problem ==
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The number of real values of <math>x</math> which satisfy the equation <cmath>(2^{6x+3})(4^{3x+6})=8^{4x+5}</cmath> is:
+
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}}   
  
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[[Category:]]
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[[Category:Introductory Algebra Problems]]
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 19:48, 24 March 2023

Problem

The number of real values of $x$ that satisfy the equation \[(2^{6x+3})(4^{3x+6})=8^{4x+5}\] is:

$\text{(A) zero} \qquad \text{(B) one} \qquad \text{(C) two} \qquad \text{(D) three} \qquad \text{(E) greater than 3}$

Solution

We know that $2^{6x+3}\cdot4^{3x+6}=2^{6x+3}\cdot(2^2)^{3x+6}=2^{6x+3}\cdot2^{6x+12}=2^{12x+15}$. We also know that $8^{4x+5}=(2^3)^{4x+5}=2^{12x+15}$. There are infinite solutions to the equation $2^{12x+15}=2^{12x+15}$, so the answer is $\boxed{\text{(E) greater than 3}}$.

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

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