The value of (2^(-14) + 2^(-15) + 2^(-16) + 2^(-17))/5 is

We start by translating the question. We are asked (2^-14) + (2^-15) + (2^-16) + (2^-17) is how times the value of 2^-17. We can express it as the following:



The answer is C.

\(2^{-14} + 2^{-15} + 2^{-16} + 2^{-17}/5\) -----> Factor out from the nominator \(2^{-17}\)
\(2^{-17}(2^3+2^2+2^1+1)/5\)
\(2^{-17}(8+4+2+1)/5\)
\(2^{-17}*15/5\)
\(2^{-17}*3\)

Therefore, the equation is \(3\) times the value of \(2^{-17}\) and our answer is

Comments please!

2^-14+2^-15+2^-16+2^-17/5

= 2^-17(2*3 + 2^2 + 2 + 1)/5

= = 2^-17 * 15/5 = 3(2^-17)

Answer - C

Why do we divide by 2^-17?

Thanks,
C

It is given in the question.

We need to find "how many times the value of 2^(-17)" which means the entire expression is divided by 2^(-17). The quotient is the answer

Hope it is clear

How many times the value of 2^{-17} means just multiply the complete equation by 2^17 (Please note the power sign has been changed) & we get the answer

(8+4+2+1) / 5 = 3 = Answer = C

The question asked
the value of (2^(-14) + 2^(-15) + 2^(-16) + 2^(-17))/5 is how many times the value of 2^(-17)?

Lets say (2^(-14) + 2^(-15) + 2^(-16) + 2^(-17))/5 = x* 2^(-17) --> (X times of 2^(-17)), So basically we need to find what is x ?

looks like we need to simplify the exponents to make given values in form desire one ..

lets Simplify Given Part , if we take 2^(-17) from numerator the simplified numerator will be ..

2^(-17)(2^(3) + 2^(2) + 2^(1) + 2^(0) )/5 = x*2^(-17)

why 2^(3) + 2^(2) ..... ? because if we want to make 2^(-17) = 2^(-14) we required to add 2^(3) positive power . same for other...

now divide 2^(-17) both side , x = (2^(3) + 2^(2) + 2^(1) + 2^(0) )/5 => x= (8+4+2+1)/5 => x =15/5 => x=3 Answer is C.

See attachment
Time: 30 Seconds

Riffing on Scott's solution above. Instead of 'pulling out' a 2^-17, you can multiply both sides by 2^17 to simplify all of the exponents.

Attached is a visual that should help.

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

To simplify multiply both sides by \(2^{17}\) to get:-

\({\frac{1}{5}*(2^3+2^2+2^1+1)}\) = K * 1[/m]

This implies k = 3 (Correct Answer)

gmatbuster - Can you please share your solution?

Hi GMATters,

Here's my video solution to this problem:



Best,

Rowan

I solved this a slightly different way (more based on reasoning);

To start, I noted that each of the terms in the numerator can be broken out separately, giving us one of the operations as 2^-17 / 5 (remembering that a + b / c = a / c + b / c).

In comparing this to the value the problem is asking us to evaluation, we can note that this specific operation (mentioned above) will represent 1/5 of the value of 2^-17.

Moving down the line, we come across our second operation 2^-16 / 5. Thinking logically, I can presume that there is one less (1/2) in this expression than in the preceding operation (2^-17 / 5) since 2^-16 can also be written as 1/2^16.

Thus, in order to compare magnitudes from the first operation we evaluated, we can multiply the original magnitude by 2, yielding us 2 / 5 of the value the problem is asking us to evaluate (1 / 5 * 2 = 2 / 5) (Note: We multiply by 2 here because for our term, 2^-16, to have one less (1/2) means that there was a (1/2) divided out of the original operation, 2^-17. Therefore, to divide out a (1/2) is the equivalent of multiplying our operation, 2^-17 by 2 to yield 2^-16).

If we continue this train of thought for each term, we'll recognize 2^-15 has TWO less (1/2)'s, therefore we multiply our magnitude by 4... and so on and so forth.

Adding up all of our terms' magnitudes in relation to the value the problem is asking us to evaluate, we end with 15 / 5 or 3.

I will admit, algebra would have been easier, but I like to think through these things.

Feel free to correct me if my logic is flawed in any of the above.

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