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m and n are positive numbers. (2^m)^n = 512. What is the min

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m and n are positive numbers. (2^m)^n = 512. What is the min

by Max@Math Revolution » Tue May 28, 2019 11:35 pm

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[GMAT math practice question]

m and n are positive numbers. (2^m)^n = 512. What is the minimum value of m + n?

A. 3
B. 4
C. 5
D. 6
E. 8
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by Max@Math Revolution » Fri May 31, 2019 12:13 am
=>

(2^m)^n = 2^{mn} = 512 = 2^9.
Thus mn = 9.
Recall that if mn is a fixed constant, and m and n are positive numbers, then m + n has its minimum value when m = n.
Thus, we must have m = n = 3 to obtain the minimum value of m + n = 3 + 3 = 6.

Therefore, the answer is D.
Answer: D
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by Scott@TargetTestPrep » Sat Jun 01, 2019 5:25 am
Max@Math Revolution wrote:[GMAT math practice question]

m and n are positive numbers. (2^m)^n = 512. What is the minimum value of m + n?

A. 3
B. 4
C. 5
D. 6
E. 8

We can rewrite the equation:

(2^m)^n = 512

2^(mn) = 2^9

mn = 9

To minimize m+n, find the values of m and n such that they are as close to each other as possible. Thus, we see that m = 3 and n = 3, and so their sum m + n = 6.

Answer: D

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by deloitte247 » Fri Jun 07, 2019 11:27 pm
$$\left(2^m\right)^n=512$$
$$2^{mn}=512$$
$$2^9=512$$
$$2^{mn}=2^9$$
$$mn=9$$
Product of 2 positive number that will yield 9 are (1 and9) or (3 and 3)
1*9 = 9
3*3 = 9
when m =1 and n = 9
m + n = 10
when m = 3 and n = 3
m + n = 6
Minimum value of m + n will occur when
m = 3 and n = 3
m + n = 6,

$$Answer\ is\ Option\ D$$
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