Difference between revisions of "1985 AHSME Problems/Problem 15"

Revision as of 23:56, 2 April 2018

Problem

If $a$ and $b$ are positive numbers such that $a^b=b^a$ and $b=9a$ , then the value of $a$ is

$\mathrm{(A)\ } 9 \qquad \mathrm{(B) \ }\frac{1}{9} \qquad \mathrm{(C) \ } \sqrt[9]{9} \qquad \mathrm{(D) \ } \sqrt[3]{9} \qquad \mathrm{(E) \ }\sqrt[4]{3}$

Solution

Substitue $b=9a$ into $a^b=b^a$ to get $a^{9a}=(9a)^a$ . Since $x^{yz}=(x^y)^z$ , we have $a^{9a}=(a^9)^a$ , and $(a^9)^a=(9a)^a$ . Taking the

$a\text{th}$ root of both sides gives $a^9=9a$ . Dividing by $a$ yields $a^8=9\implies a=\sqrt[8]{9}=\sqrt[4]{3}, \boxed{\text{E}}$ .

1985 AHSME (Problems • Answer Key • Resources)
Preceded by Problem 14	Followed by Problem 16
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All AHSME Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

@@ Line 1: / Line 1: @@
 ==Problem==
-If <math> a </math> and <math> b </math> are positive numbers such that <math> a^b=b^a </math> and <math> b=9a </math>, then the value of <math> a </math> is:
+If <math> a </math> and <math> b </math> are positive numbers such that <math> a^b=b^a </math> and <math> b=9a </math>, then the value of <math> a </math> is
 <math> \mathrm{(A)\ } 9 \qquad \mathrm{(B) \ }\frac{1}{9} \qquad \mathrm{(C) \  } \sqrt[9]{9} \qquad \mathrm{(D) \  } \sqrt[3]{9} \qquad \mathrm{(E) \  }\sqrt[4]{3} </math>

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Difference between revisions of "1985 AHSME Problems/Problem 15"

Revision as of 23:56, 2 April 2018

Problem

Solution

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