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. | ||
− | The LHS is <math>(a^b | + | 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 21:34, 13 July 2019
Problem
Given the binary operation defined by
for all positive numbers
and
. Then for all positive
, we have
Solution
Let . If all of them are false, the answer must be
. If one does not fail, we will try to prove it.
For option , we have
, which is clearly false.
For option , we have
, which is false.
For option , we have
, which is false.
For option , we have
, which is true.
The LHS is . By the elementary definition of exponentiation, this is
multiplied by itself
times. Since each
is actually
multiplied
times, the expression
is
multiplied by itself
times.
The RHS is . This is
multiplied by itself
times.
Thus, the LHS is always equal to the RHS, so 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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.