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18 Apr 2022, 00:00
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D
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
45%
(medium)
Question Stats:
72% (02:17) correct
28%
(02:52)
wrong
based on 749
sessions
History
Date
Time
Result
Not Attempted Yet
If \(6^{2x-2} = 8100\), then what is the value of \(6^x\)?
(A) 90
(B) 135
(C) 270
(D) 540
(E) 1080
(A) 90
(B) 135
(C) 270
(D) 540
(E) 1080
18 Apr 2022, 07:24
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GMATNinja
Given: \(6^{2x-2} = 8100\)
Rewrite both sides as follows: \((6^{x-1})^2 = 90^2\)
Take the square root of both sides to get: \(6^{x-1} = 90\)
Multiply both sides of the equation by \(6\) (aka \(6^1\)) to get: \((6^{x-1})(6^1) = (90)(6^1)\)
Simplify: \(6^x = 540\)
General Discussion
18 Apr 2022, 09:04
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Breaking down 8100 by prime factorisation, 8100 = 81 * 100 = \(3^4 * 2^2 * 5^2\).
Using the laws of exponents, \(6^{2x-2}\) = \(\frac{6^{2x} }{ 6^2}\) = \(\frac{(6^x)^2 }{ 6^2}\).
Equating the two sides, \(\frac{(6^x )^2}{6^2}\) = \(3^4 * 2^2 * 5^2\)
Since 6 = 2 * 3, \(6^2\) = \(2^2 * 3^2\). Cross multiplying the denominator of the LHS with the RHS, we have,
\((6^x)^2\) = \(2^4 * 3^6 * 5^2\)
Taking the square root on both sides, \(6^x\) = \(2^2 * 3^3\) * 5 = 4 * 27 * 5 = 540.
The correct answer option is D.
Using the laws of exponents, \(6^{2x-2}\) = \(\frac{6^{2x} }{ 6^2}\) = \(\frac{(6^x)^2 }{ 6^2}\).
Equating the two sides, \(\frac{(6^x )^2}{6^2}\) = \(3^4 * 2^2 * 5^2\)
Since 6 = 2 * 3, \(6^2\) = \(2^2 * 3^2\). Cross multiplying the denominator of the LHS with the RHS, we have,
\((6^x)^2\) = \(2^4 * 3^6 * 5^2\)
Taking the square root on both sides, \(6^x\) = \(2^2 * 3^3\) * 5 = 4 * 27 * 5 = 540.
The correct answer option is D.
