The Value Of (2^(-14) + 2^(-15) + 2^(-16) + 2^(-17))/5 Is GMAT Problem Solving

Question: The value of \(\frac{2^{(-14)}+2^{(-15)}+2^{(-16)}+2^{(-17)}}{5}\) is how many times the value of \(2^{(-17)} \)?

3/2
5/2
3
4
5

“The value of (2^(-14) + 2^(-15) + 2^(-16) + 2^(-17))/5 is” - is a topic that is covered in the quantitative reasoning section of the GMAT. To successfully execute the GMAT Problem Solving questions, a student must possess a wide range of qualitative skills. The entire GMAT Quant section consists of 31 questions. The problem-solving section of the GMAT Quant topics requires the solution of calculative mathematical problems.

Solutions and Explanation

Approach Solution: 1

Let us find the value of \(\frac{\frac{1}{5}* (2^{(-14)} + 2^{(-15)} + 2^{(-16)}+2^{(-17)})}{2^{(-17)}}\)=\(\frac{\frac{1}{5}*(\frac{1}{_2{14}}+\frac{1}{_2{15}}+\frac{1}{_2{16}}+\frac{1}{_2{17}})}{\frac{1}{_2{17}}}\)

We get,\(\frac{\frac{1}{5}*(\frac{1}{_2{14}}+\frac{1}{_2{15}}+\frac{1}{_2{16}}+\frac{1}{_2{17}})}{\frac{1}{_2{17}}}\)

\(=\frac{2^{17}}{5}*+(\frac{1}{2^{14}}+\frac{1}{2^{15}}+\frac{1}{2^{16}}+\frac{1}{2^{17}})\)

\(=\frac{1}{5}*(2^3+2^2+2+1)=\frac{1}{5}*15=3\)

Correct Answer: (C)

Approach Solution : 2

Lets say (2^(-14) + 2^(-15) + 2^(-16) + 2^(-17))/5 = x* 2^(-17) where (X times of 2^(-17)),
So we need to find the value of x
It appears that we must simplify the exponents in order to convert the given values into the desired form.

Let's Simplify the Given Part ;
2^(-17) is taken out from the Numerator and the Simplified numerator will be
2^(-17)(2^(3) + 2^(2) + 2^(1) + 2^(0) )/5 = x*2^(-17)
You might be wondering why 2^(3) + 2^(2) ..... is used.
This is due to the fact that we must add 2(3) positive power to make 2^(-17) = 2^(-14). This also applies to other

Now divide 2^(-17) on both sides ,
x = (2^(3) + 2^(2) + 2^(1) + 2^(0) )/5
x= (8+4+2+1)/5
x =15/5

Finally we get, x=3.

Correct Answer: (C)

Approach Solution : 3

The nominator 2^(-17) has to be factored out from the following, 2^(-14) + 2^(-15) + 2^(-16) + 2^(-17))/5

=>2^(-17) * { [(2^3) + (2^2) + 2 + 1] / 5 }
=>2^(-17) * { (8 + 4 + 2 + 1) / 5 }
=>2^(-17) * 3

Therefore 3 is the answer.

Correct Answer: (C)

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