Problem Solving

[GMAT math practice question]

If 272/4^6 = 1/4^m+ 1/4^n, what is the value of mn?

A. 2
B. 4
C. 6
D. 8
E. 12

=>

$$\frac{272}{4^6}=\frac{17}{4^4}=\frac{1+16}{4^4}=\frac{1}{4^4}+\frac{16}{4^4}=\frac{1}{4^4}+\frac{1}{4^2}$$

Thus m = 4 and n = 2, or m = 2 and n = 4.
So, mn = 8.

Therefore, the answer is D.
Answer: D

$$If\ \frac{272}{4^6}=\frac{1}{4^m}+\frac{1}{4^n}\ what\ is\ mn$$
$$\frac{\left(4^{^2}\cdot17\right)}{4^6}=\frac{1}{4^m}\cdot\frac{1}{4^n}$$
$$\frac{\left(4^2\cdot17\right)}{4^2\cdot4^4}=\frac{1}{4^m}\cdot\frac{1}{4^n}$$
$$\frac{17}{4^4}=\frac{1}{4^m}\cdot\frac{1}{4^n}$$
$$\frac{4^2+1}{4^4}=\frac{1}{4^m}+\frac{1}{4^n}$$
$$\frac{4^2}{4^4}+\frac{4^2}{4^4}=\frac{1}{4^m}+\frac{1}{4^n}$$
in terms of power
$$\frac{4^2}{4^4}+\frac{1^1}{4^4}=\frac{1^1}{4^2}+\frac{1^1}{4^4}$$
$$\frac{1}{4^2}+\frac{1}{4^4}=\frac{1}{4^m}+\frac{1}{4^n}$$
Comparing the two side values of m and n = 2 and 4 respectively.
Therefore, the product is mn = 2*4 = 8

$$Answer\ is\ Option\ D$$