Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "1970 AHSME Problems/Problem 13"

(Created page with "== Problem == Given the binary operation <math>\star</math> defined by <math>a\star b=a^b</math> for all positive numbers <math>a</math> and <math>b</math>. Then for all positiv...")
 
(→‎Solution)
 
(2 intermediate revisions by 2 users not shown)
Line 10: Line 10:
  
 
== 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.
 +
 
 +
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.
 +
 
 +
Thus, the LHS is always equal to the RHS, so <math>\fbox{D}</math> is the only correct statement.
  
 
== See also ==
 
== See also ==
−
{{AHSME box|year=1970|num-b=12|num-a=14}}   
+
{{AHSME 35p box|year=1970|num-b=12|num-a=14}}   
  
 
[[Category: Introductory Algebra Problems]]
 
[[Category: Introductory Algebra Problems]]
 
{{MAA Notice}}
 
{{MAA Notice}}

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

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

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=1970_AHSME_Problems/Problem_13&oldid=107612"