Difference between revisions of "1961 AHSME Problems/Problem 9"
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| − | Let <math>r</math> be the result of doubling both the base and | + | == Problem 9== |
| + | |||
| + | Let <math>r</math> be the result of doubling both the base and exponent of <math>a^b</math>, and <math>b</math> does not equal to <math>0</math>. | ||
| + | If <math>r</math> equals the product of <math>a^b</math> by <math>x^b</math>, then <math>x</math> equals: | ||
| + | |||
| + | <math>\textbf{(A)}\ a \qquad | ||
| + | \textbf{(B)}\ 2a \qquad | ||
| + | \textbf{(C)}\ 4a \qquad | ||
| + | \textbf{(D)}\ 2\qquad | ||
| + | \textbf{(E)}\ 4 </math> | ||
| + | |||
| + | ==Solution== | ||
| + | From the problem, <math>r = (2a)^{2b}</math>, so | ||
| + | <cmath>(2a)^{2b} = a^b \cdot x^b</cmath> | ||
| + | <cmath>(4a^2)^b = (ax)^b</cmath> | ||
| + | <cmath>4a^2 = ax</cmath> | ||
| + | <cmath>x = 4a</cmath> | ||
| + | Thus, the answer is <math>\boxed{\textbf{(C)}}</math>. | ||
| + | |||
| + | ==See Also== | ||
| + | {{AHSME 40p box|year=1961|num-b=8|num-a=10}} | ||
| + | |||
| + | {{MAA Notice}} | ||
| + | |||
| + | [[Category:Introductory Algebra Problems]] | ||
Latest revision as of 13:36, 19 May 2018
Problem 9
Let 
be the result of doubling both the base and exponent of 
, and 
does not equal to 
.
If equals the product of by 
, then 
equals:
Solution
From the problem, 
, so
![\[(2a)^{2b} = a^b \cdot x^b\]](http://latex.artofproblemsolving.com/7/5/a/75a10b5a21e23f08bab3874b295fc1479a749ea8.png)
![\[(4a^2)^b = (ax)^b\]](http://latex.artofproblemsolving.com/1/c/5/1c5ad8b4a949d81c868aa2b52980c41be9cd3277.png)
![\[4a^2 = ax\]](http://latex.artofproblemsolving.com/0/2/3/0238badcc2410594bab9da515d4d4d32b79148f5.png)
![\[x = 4a\]](http://latex.artofproblemsolving.com/2/b/a/2baf190c183b951cd39eb54e1294ea51e6d7ac59.png)
Thus, the answer is 
.
See Also
| 1961 AHSC (Problems • Answer Key • Resources) | ||
| Preceded by Problem 8 |
Followed by Problem 10 | |
| 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 • 36 • 37 • 38 • 39 • 40 | ||
| All AHSME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
