Difference between revisions of "1970 AHSME Problems/Problem 13"
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== Solution == | == Solution == | ||
− | <math>\fbox{D}</math> | + | Let <math>a = 2, b = 3, c=n = 4</math>. If all of them are false, the answer must be <math>E</math>. If one does not fail, we will try to prove it. |
+ | |||
+ | For option <math>A</math>, we have <math>2^3 = 3^2</math>, which is clearly false. | ||
+ | |||
+ | For option <math>B</math>, we have <math>2^{81} = 8^{4}</math>, which is false. | ||
+ | |||
+ | For option <math>C</math>, we have <math>2^{81} = 16^3</math>, which is false. | ||
+ | |||
+ | 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. | ||
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+ | The RHS is <math>a^{bn}</math>. This is <math>a</math> multiplied by itself <math>bn</math> times. | ||
+ | |||
+ | Thus, the LHS is always equal to the RHS, so <math>\fbox{D}</math> is the only correct statement. | ||
== See also == | == See also == |
Revision as of 20:33, 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 $(a^b}^n$ (Error compiling LaTeX. ! Extra }, or forgotten $.). By the elementary definition of exponentiation, this is multiplied by itself times. Since each is actually multiplied times, the expression $(a^b}^n$ (Error compiling LaTeX. ! Extra }, or forgotten $.) 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.