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Difference between revisions of "1970 AHSME Problems/Problem 13"
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For option <math>D</math>, we have <math>8^4 = 2^{12}</math>, which is true.
 
For option <math>D</math>, we have <math>8^4 = 2^{12}</math>, which is true.
  
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The LHS is <math>(a^b}^n</math>.  By the elementary definition of exponentiation, this is <math>a^b</math> multiplied by itself <math>n</math> times.  Since each <math>a^b</math> is actually <math>a</math> multiplied <math>b</math> times, the expression <math>(a^b}^n</math> is <math>a</math> multiplied by itself <math>bn</math> times.   
+
The LHS is <math>(a^b)^n</math>.  By the elementary definition of exponentiation, this is <math>a^b</math> multiplied by itself <math>n</math> times.  Since each <math>a^b</math> is actually <math>a</math> multiplied <math>b</math> times, the expression <math>(a^b)^n</math> is <math>a</math> multiplied by itself <math>bn</math> times.   
  
 
The RHS is <math>a^{bn}</math>.  This is <math>a</math> multiplied by itself <math>bn</math> times.
 
The RHS is <math>a^{bn}</math>.  This is <math>a</math> multiplied by itself <math>bn</math> times.

Latest revision as of 20:34, 13 July 2019

Problem

Given the binary operation $\star$ defined by $a\star b=a^b$ for all positive numbers $a$ and $b$. Then for all positive $a,b,c,n$, we have

$\text{(A) } a\star b=b\star a\quad\qquad\qquad\ \text{(B) } a\star (b\star c)=(a\star b) \star c\quad\\ \text{(C) } (a\star b^n)=(a \star n) \star b\quad \text{(D) } (a\star b)^n =a\star (bn)\quad\\ \text{(E) None of these}$

Solution

Let $a = 2, b = 3, c=n = 4$. If all of them are false, the answer must be $E$. If one does not fail, we will try to prove it.

For option $A$, we have $2^3 = 3^2$, which is clearly false.

For option $B$, we have $2^{81} = 8^{4}$, which is false.

For option $C$, we have $2^{81} = 16^3$, which is false.

For option $D$, we have $8^4 = 2^{12}$, which is true.

The LHS is $(a^b)^n$. By the elementary definition of exponentiation, this is $a^b$ multiplied by itself $n$ times. Since each $a^b$ is actually $a$ multiplied $b$ times, the expression $(a^b)^n$ is $a$ multiplied by itself $bn$ times.

The RHS is $a^{bn}$. This is $a$ multiplied by itself $bn$ times.

Thus, the LHS is always equal to the RHS, so $\fbox{D}$ is the only correct statement.

See also

1970 AHSC (Problems • Answer Key • Resources)
Preceded by
Problem 12 		Followed by
Problem 14
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 • 31 • 32 • 33 • 34 • 35
All AHSME Problems and Solutions

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