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If 5^11 × 4^5 = 5 × 10^k , what is the value of k?

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by Brent@GMATPrepNow » Wed Sep 04, 2013 3:49 pm
soneee wrote:I am having trouble with this question.

If (5^11)(4^5) = 5(10^k) , what is the value of k?
(5^11)(4^5) = 5(10^k)
(5^11)((2²)^5) = 5(10^k) [replaced 4 with 2²]
(5^11)(2^10) = 5(10^k) [applied power of a power law]
(5^1)(5^10)(2^10) = 5(10^k) [rewrote 5^11 as (5^1)(5^10)]
(5^1)(10^10) = 5(10^k) [combined bases since we had same powers]
So, we can see that [spoiler]k = 10[/spoiler]

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by GMATGuruNY » Wed Sep 04, 2013 9:05 pm
soneee wrote:I am having trouble with this question.

If 5^11 × 4^5 = 5 × 10^k , what is the value of k?
We can solve for k by focusing on the number of 5's on each side of the equation.

5¹¹ * 4� = 5¹ * 10^k.
5¹¹ * 4� = 5¹ * 5^k * 2^k.
Since there are eleven 5's on the lefthand side, there must also be eleven 5's on the righthand side, implying that k=10.
5¹¹ * 4� = 5¹ * 5¹� * 2^k.
5¹¹ * 4� = 5¹¹ * 2^k.

The correct answer is k=10.
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