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Math Help - GMAT question

  1. #1
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    GMAT question

    Hi everyone,

    So for several months I have been busy studying for the GMAT which I'll be taking this coming Saturday. Today I took a prep. test and I am just blown away by how hard it is. The prep test is much harder than the math problems that I solved in the books I have been studying. I was wondering if you could help me solve a couple (or more) of these problems that are giving me a really hard time so that I could be as prepared as I possibly can. Thanks in advance for any help.

    Q: If 5^21 x 4^11 = 2 x 10^n, then what is n?

    Just in case...the ^ symbol represents exponent. Using a calculator I can work this out, but in the test I can't use a calculator. Anyone know of a shortcut? Thanks.
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  2. #2
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    Quote Originally Posted by crazirani View Post

    Q: If 5^21 x 4^11 = 2 x 10^n, then what is n?

    Just in case...the ^ symbol represents exponent. Using a calculator I can work this out, but in the test I can't use a calculator. Anyone know of a shortcut? Thanks.
    Prime factorize,
    4^{11}=(2^2)^{11}=2^{22}
    And,
    10^n=(2\cdot 5)^n=2^n\cdot 5^n.
    Thus,
    5^{21}\times 4^{11}=5^{21}\times 2^{22}
    And,
    2\times 10^n=2\times (2\cdot 5)^n=2\times 2^n \times 5^n=2^{n+1}\cdot 5^n
    Thus,
    5^{21}\times 2^{22}=5^n \cdot 2^{n+1}
    By just looking at the exponents we see that,
    n=21
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  3. #3
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    That's amazing! Thanks a lot.

    EDIT: I just realized that I posted this in the "Elementary and Middleschool" folder...sorry.
    Last edited by crazirani; January 17th 2007 at 12:39 PM.
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  4. #4
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    Quote Originally Posted by crazirani View Post
    Hi everyone,

    So for several months I have been busy studying for the GMAT which I'll be taking this coming Saturday. Today I took a prep. test and I am just blown away by how hard it is. The prep test is much harder than the math problems that I solved in the books I have been studying. I was wondering if you could help me solve a couple (or more) of these problems that are giving me a really hard time so that I could be as prepared as I possibly can. Thanks in advance for any help.

    Q: If 5^21 x 4^11 = 2 x 10^n, then what is n?

    Just in case...the ^ symbol represents exponent. Using a calculator I can work this out, but in the test I can't use a calculator. Anyone know of a shortcut? Thanks.
    Here is one way.

    (5^21)(4^11) = 2(10^n)
    There's a 2 at the RHS and a 4 at the LHS, so express the 4 in terms of 2,
    (5^21)(2^22) = 2(10^n)
    Divide both sides by 2,
    (5^21)(2^21) = 10^n
    (5*2)^21 = 10^n
    10^21 = 10^n
    n = 21 --------------answer.
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