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Math Help - Indices Algebra

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    Indices Algebra

    I have two questions that needs to be solved with algebra, please help, i figured out the answers to be -4 for the first and -1 for the second but i dont know the steps solving it with algebra.

    2x = 1/16 Find X
    (0.5)x = 2 Find X
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    Re: Indices Algebra

    Quote Originally Posted by TheElect View Post
    I have two questions that needs to be solved with algebra, please help, i figured out the answers to be -4 for the first and -1 for the second but i dont know the steps solving it with algebra.

    2x = 1/16 Find X
    (0.5)x = 2 Find X
    \displaystyle \begin{align*} 2^x &= \frac{1}{16} \\ 2^x &= \frac{1}{2^4} \\ 2^x &= 2^{-4} \\ x&= 4 \\ \\ \left(0.5\right)^x &= 2 \\ \left(\frac{1}{2}\right)^x &= 2 \\ \frac{1}{2^x} &= 2 \\ 2^{-x} &= 2^1 \\ -x &= 1 \\ x &= -1 \end{align*}
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    Re: Indices Algebra

    Thankyou, but can you please explain further into how that works? Sorry, my understanding of indices aren't that good
    Last edited by TheElect; July 17th 2012 at 04:46 AM.
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    Re: Indices Algebra

    Quote Originally Posted by TheElect View Post
    Thankyou, but can you please explain further into how that works? Sorry, my understanding of indices aren't that good
    That was every step. I don't know what else I can show you...
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    Re: Indices Algebra

    Your problem seems to be not understanding what "indices" (I would say "exponents" which is a little more specific than "indices") mean. x^2 ('x squared') means 'x times x', x^3 means 'x times x times x', etc. Further, a negative exponent means "reciprocal" or "1 over" the value. In particular 2^4= 2(2)(2)(2)= 16 so that 2^{-4}= \frac{1}{16}.

    Of course, 0.5= \frac{1}{2} so that 0.5= \left(\frac{1}{2^1}\right)= 2^{-1} and (0.5)^x= (2^{-1})^x= 2^{-x}

    So the first equation says 2^x= 2^{-4} and comparing the two sides x= -4.

    The second equation says 2^{-x}= 2^1 and comparing the two sides gives -x= 1.

    (There is a technical point here- that the exponential function is "one-to-one" which means that if f(x)= f(y) then x= y. Not all functions have that property but the exponential does. All you need to know is that you can do that for exponential functions.)
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    Re: Indices Algebra

    4^x = 4^y
    Can you "see" that x=y?
    Let x=2: 4^2 = 16; so 4^y = 16; so y HAS TO equal x : capish?
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    Re: Indices Algebra

    I think I understand now, thanks all of you
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    Re: Indices Algebra

    Definition of exponentiation: b^m = b * b * b * * b (b times itself m times)
    For example: b^4 b^3 =
    ( b b b b) ( b b b) = (definition of exponentiation)
    b b b b b b b = (associative property of multiplication)
    b^7 (definition of exponentiation)
    If b^m=b^n
    b^m/b^n=1
    b^(m-n)=1
    m-n=0
    Hence, m=n
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