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Official Guide to the GMAT 2017 Data Sufficiency #277

Jonathan EvansJonathan Evans PowerScore StaffJoined: 10/31/2016Posts: 109
A student asks:

I have a question about question #277’s explanation in the GMAT official guide, page 309.  It states as the formula for determining the answer to the question as whether “1 is greater than 10^2n,” and I don’t understand how they came up with that equation, although the main determination being whether or not n is negative does make sense to me mostly.  

 

That said,  I’d also like to know if the reason n being negative is crucial is due to the fact that in the case of a fraction with a denominator raised to a negative exponent that causes the fraction to flip, making (0.1)^n 10 and 10^n 1/10^n.  Or would (0.1)^n turn into a 1 / 1/10^n?  Could be crucial regardless I think but wanted to know how that works.  


Response below.

Posts

  • Jonathan EvansJonathan Evans PowerScore Staff Joined: 10/31/2016Posts: 109
    Here's what's happening in the explanation. The authors multiply both sides by 10 to eliminate the decimal in the first value. The question becomes:

    Is 1ⁿ greater than 10²ⁿ?

    Note then that if 1 is raised to any power, it remains 1. Thus, note the possibilities:

    • 1 > 10²ⁿ iff n < 0. This follows because given that n is negative, the power will be a fraction smaller than 1.
    Supply a number to illustrate. n = -1. Then we have 10⁻² or 1/10².
    • 1 = 10²ⁿ iff n = 0. This follows because given that n is 0, the power will be equal to 1.
    Supply n = 0. Then we have 10⁰ or 1.
    • 1 < 10²ⁿ iff n > 0. This follows because given that n is greater than 0, the exponent 2n will also be greater than 0. Any number supplied for n, even an infinitesimal fraction, will still result in some power greater than 1.
    For instance, supply n = 1. Then we have 10² or 100.

    Thus, given that neither statement makes a distinction of whether n must be greater than, less than, or equal to 0, neither is sufficient by itself or along with the other. E is the credited response.

    There are of course other approaches.

    Alternatively, start again with the initial question:
    Is (0.1)ⁿ greater than 10ⁿ?
    Note that it is difficult to compare powers with different bases. What might we want to do? Try to get the same base. Note that we appear to be dealing with powers of 10. The first term, (0.1)ⁿ can be written as (1/10)ⁿ which equals (10⁻¹)ⁿ.

    When you take the power of a power, you multiply the exponents together. Thus you get 10⁻ⁿ.

    Now the question becomes:
    Is 10⁻ⁿ greater than 10ⁿ?
    Clearly here we could dispense with the base altogether and ask the question:
    Is -n greater than n?

    When is -n greater than n? -n is greater than n only when n is negative. -n equals n when n is 0. -n is less than n when n is greater than 0.

    • -n > n iff n < 0
    • -n = n iff n = 0
    • -n < n iff n > 0 
    We still arrive at E. 

    Alternatively, you could consider trying out different values for n consistent with the statements to illustrate that they are insufficient. 

    Finally, you could observe that whether the exponent is greater than or less than 10 or -10 is a completely arbitrary value for the exponent given base 10 powers and thus unlikely to give any definitive response to our question, even when considered together. 

    I hope this helps!
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