Difference between revisions of "2006 AMC 10A Problems/Problem 6"

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== Problem ==
 
== Problem ==
What non-zero real value for <math>\displaystyle x</math> satisfies <math>\displaystyle(7x)^{14}=(14x)^7</math>?
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What non-zero real value for <math>x</math> satisfies <math>(7x)^{14}=(14x)^7</math>?
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<math> \textbf{(A) } \frac17\qquad \textbf{(B) } \frac27\qquad \textbf{(C) } 1\qquad \textbf{(D) } 7\qquad \textbf{(E) } 14 </math>
  
<math> \mathrm{(A) \ } \frac17\qquad \mathrm{(B) \ } \frac27\qquad \mathrm{(C) \ } 1\qquad \mathrm{(D) \ } 7\qquad \mathrm{(E) \ } 14 </math>
 
 
== Solution ==
 
== Solution ==
We first break up 14 into (7x)(2), so that
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Taking the seventh root of both sides, we get <math>(7x)^2=14x</math>.
 
 
<math>(7x)^{14}=(7x\cdot 2)^7\Longrightarrow (7x)^{14}=((7x)^7)(2^7)</math>
 
 
 
We then divide out <math>(7x)^7</math>
 
 
 
<math>\frac{(7x)^{14}}{(7x)^7}=\frac{((7x)^7)(2^7)}{(7x)^7}</math>
 
  
<math>(7x)^7=2^7</math>
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Simplifying the LHS gives <math>49x^2=14x</math>, which then simplifies to <math>7x=2</math>.
  
We take the 7th root of each side.
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Thus, <math>x=\boxed{\textbf{(B) }\frac{2}{7}}</math>.
  
<math>\sqrt[7]{(7x)^7}=\sqrt[7]{2^7}</math>
 
 
<math>7x=2\Longrightarrow x=\frac{2}{7}, (B)</math>
 
 
== See also ==
 
== See also ==
 
{{AMC10 box|year=2006|ab=A|num-b=5|num-a=7}}
 
{{AMC10 box|year=2006|ab=A|num-b=5|num-a=7}}
 
*[[2006 AMC 10A Problems]]
 
  
 
[[Category:Introductory Algebra Problems]]
 
[[Category:Introductory Algebra Problems]]
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{{MAA Notice}}

Latest revision as of 13:28, 31 May 2023

Problem

What non-zero real value for $x$ satisfies $(7x)^{14}=(14x)^7$?

$\textbf{(A) } \frac17\qquad \textbf{(B) } \frac27\qquad \textbf{(C) } 1\qquad \textbf{(D) } 7\qquad \textbf{(E) } 14$

Solution

Taking the seventh root of both sides, we get $(7x)^2=14x$.

Simplifying the LHS gives $49x^2=14x$, which then simplifies to $7x=2$.

Thus, $x=\boxed{\textbf{(B) }\frac{2}{7}}$.

See also

2006 AMC 10A (ProblemsAnswer KeyResources)
Preceded by
Problem 5
Followed by
Problem 7
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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