$2^{(25)} + 2^{(26)} + 2^{(27)} + ... + 2^{(46)} + 2^{(47)} =$

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2^{25}(2^{(23)}-1)

Correct

2^{25}(2^{(27)}-1)

2^{25}(2^{(28)}-1)

2^{26}(2^{(24)}-1)

2^{26}(2^{(25)}-1)

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Explanation

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Official Explanation

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This is a great question! To solve it, we can apply the Three Guiding Principles for Exponents: 1) Identify common bases, 2) Transform addition into multiplication, and 3) Recognize patterns. In this case, we have the terms from 2^{25} to 2^{47}. To simplify, let's first consider a smaller example: 2^{25} + 2^{26} + 2^{27} + 2^{28}. We can factor out the common base of 2^{25}: 2^{25}(1 + 2^1 + 2^2 + 2^3) = 2^{25}(1 + 2 + 4 + 8) = 2^{25}(15). We can express 15 as 16 - 1, or 2^4 - 1, leading…

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Comments

WilsonDec 19, 2024

thanks for the answer

What this question tests

What it tests

Your grasp of exponent rules — multiplying and dividing powers, power-of-a-power, and negative/zero exponents.

Common trap

Adding exponents when multiplying bases that have the same exponent, or confusing (x^a)^b with x^(a·b).

2(25)+2(26)+2(27)+...+2(46)+2(47)=2^{(25)} + 2^{(26)} + 2^{(27)} + ... + 2^{(46)} + 2^{(47)} =2(25)+2(26)+2(27)+...+2(46)+2(47)=