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Exasperating Exponents

2^1 + 2^2 + 2^3 + 2^4 + 2^5

(A) 2^14

(B) 2^15

(C) 2^9 + 2^5

(D) 2^7 – 2

(E) 2^6 – 2

In this problem many students are tempted to simply add the exponents together and choose answer choice (B).

2^15, however, is not the answer. Were we multiplying the 2’s together then we would add the exponents. The general rule is:

if the bases are the same then add the exponents, only if you are multiplying

In this case, the bases — the number 2 — are the same. So what do we if the bases are being added, not multiplied?

There is no general rule here if you are adding the same bases with different exponents. This problem actually requires that you identify a pattern. When looking for a pattern we want to start with the lower numbers.

2^1 + 2^2 = 6

2^1 + 2^2 + 2^3 = 14

2^1 + 2^2 + 2^3 + 2^4 = 30

You may notice that the sum of each of the series above is very close to the next 2 added:

2^1 + 2^2 + 2^3 = 2^4 – 2

Continuing this pattern we get:

2^1 + 2^2 + 2^3 + 2^4 = 2^5 – 2

So for the original problem the greatest exponent we are adding is 2^5. Therefore, the answer to the original question is 2^6 – 2. Answer choice (E).

 

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