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Math Help - [SOLVED] Solve for X

  1. #1
    mathman619
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    [SOLVED] Solve for X

    Solve For X

    2^(2/3)x+1-3*2^(1/3)x-20=0

    ^ means to the power of

    -3 and -20 are separate from the exponents
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  2. #2
    MHF Contributor
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    Quote Originally Posted by mathman619
    Solve For X

    2^(2/3)x+1-3*2^(1/3)x-20=0

    ^ means to the power of

    -3 and -20 are separate from the exponents
    If that is
    2^[(2/3)(x+1)] -3*2^[(1/3)x] -20 = 0, ---------(i)
    then,

    2^[(2/3)x] *2^(2/3) -2^[(1/3)x] *3 -20 = 0 ----(ii)
    Let y = 2^[(1/3)x]
    Then, (ii) becomes
    [2^(2/3)]y^2 -3y -20 = 0
    Use the Quadratic Formula,
    y = {-(-3) +,-sqrt[(-3)^2 -4(2^(2/3))(-20)]} / [2*2^(2/3)]
    y = {3 +,-sqrt(135.9920842)} /(3.174802104)
    y = {3 +,-11.6615644}/(3.174802104)
    y = 4.618 or -2.728

    Umm, 0.618.....

    When y = 4.618,
    2^[(1/3)x] = 4.618
    2^(x/3) = 4.618
    Take the common log of both sides,
    (x/3)log(2) = log(4.618)
    x/3 = log(4.618)/log(2)
    x = 3log(4.618)/log(2)
    x = 6.6218 --------------***

    When y = -2.728,
    2^(x/3) = -2.728
    log(2^(x/3)) = log(-2.728)
    Cannot be. There is no logarithm of negative numbers.

    Check x=6.6218 against the original equation,
    2^[(2/3)(x+1)] -3*2^[(1/3)x] -20 = 0 ---------(i)
    2^[(2/3)(6.6218 +1)] -3*2^[(1/3)(6.6218)] -20 =? 0
    2^(5.0812) -3*2^(2.2072667) -20 =? 0
    33.853 -13.854 -20 =? 0
    -0.001 =? 0
    Yes (that -0.001 is due to rounding of the decimals), so, OK.

    Therefore, x = 6.6218 -----------answer.
    Last edited by ticbol; March 10th 2006 at 10:50 AM. Reason: ^(x/3), not ^(2/3)
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  3. #3
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    earboth's Avatar
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    Quote Originally Posted by mathman619
    Solve For X

    2^(2/3)x+1-3*2^(1/3)x-20=0
    ^ means to the power of
    -3 and -20 are separate from the exponents
    Hello,

    only in case you mean:
    2^{\frac{2}{3} x+1}-3\cdot 2^{\frac{2}{3}x} -20 = 0

    then do the substitution as ticbol has told you. You'll get:
    y=2^{\frac{1}{3}x} . So you have:
    2 \cdot y^2 - 3 \cdot y -20=0

    Solve for y and you'get: y = 4 or y = -(5/2).

    Now you've to re-substitute
    4=2^{\frac{1}{3}x}\ \Leftrightarrow \ 2^2=2^{\frac{1}{3}x}\ \Rightarrow \ x=6

    The negative value for y is not possible with your equation, because with the positive base of 2 you'll never get a negative result.

    So the only possible result is x = 6.

    Greetings.

    EB
    Last edited by earboth; March 10th 2006 at 11:18 AM.
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