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Math Help - Calculate power of negative decimal number

  1. #1
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    Smile Calculate power of negative decimal number

    Hello
    I am new in this forum
    I try to calculate

    {-10}^{ 2.6}
    using the calculator I get an error
    but i write that way
    {-10}^{ 2.6} = {-10}^{ 2+ 6/10} = {-10}^{ 2+ 3/5} = {-10}^{2 } * {-10}^{3/5 }
    = 100 * {{-10}^{ 3} }^{1/ 5} = 100*{-1000}^{ 1/5} \approx -398.107
    i don't know if it's correct
    so for {-10}^{ 2.8} i get 630.957344480193
    Thanks in advance
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  2. #2
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    Quote Originally Posted by shayw View Post
    Hello
    I am new in this forum
    I try to calculate

    {-10}^{ 2.6}
    using the calculator I get an error
    but i write that way
    {-10}^{ 2.6} = {-10}^{ 2+ 6/10} = {-10}^{ 2+ 3/5} = {-10}^{2 } * {-10}^{3/5 }
    = 100 * {{-10}^{ 3} }^{1/ 5} = 100*{-1000}^{ 1/5} \approx -398.107
    i don't know if it's correct
    so for {-10}^{ 2.8} i get 630.957344480193
    Thanks in advance
    Your calculations are OK.

    If the exponent of the root is odd you don't get a real result.

    EDIT: Changed one word in the last sentence so it is (better?) understandable.
    Last edited by earboth; May 11th 2011 at 11:02 PM.
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  3. #3
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    Looks good to me.
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  4. #4
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    calculate power of negative decimal number

    Hi shayw,
    I assume you mean the power of a negative integer. You can simplify calculations by factoring the negative integer

    -10^2.8 =(-1)^2.8 * 10^2.8
    10^2.8 = 630.957
    (-1)^2.8 =(-1)^2 *(-1)^.8=1*1 answer is positive

    If the original problem was(-10)^2.7

    (-10)^2.7 =(-1)^2.7 * 10^2.7
    = (-1)^2 *(-1)^.5 * (-1)^.2 * 10^2.7
    = 1*-1*1=-1

    answer will be negative






    bjh
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  5. #5
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    Quote Originally Posted by earboth View Post
    Your calculations are OK.

    If the exponent of the root isn't odd you don't get a real result.
    How can you tell if a decimal number is even or odd? Do you just look at the last digit? Please realize that 10.7 also equals 10.70 and 10.700, etc. Note that 70 (and 700, 7000, ...) is even, so is 10.70 now even?
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  6. #6
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    The exponent of the root? I think you mean the exponent of the base.
    Steven
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  7. #7
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    Quote Originally Posted by mathprofessor View Post
    How can you tell if a decimal number is even or odd? Do you just look at the last digit? Please realize that 10.7 also equals 10.70 and 10.700, etc. Note that 70 (and 700, 7000, ...) is even, so is 10.70 now even?
    1. You know that

    r^{\frac ew} = \sqrt[w]{r^e}

    (In German the number w is called exponent of the root. Maybe that's the reason why I used a wrong expression)

    2. Re-write an exponent as a completely simplified fraction. Then the denominator of this fraction indicates if it is an odd or even root.

    (-10)^{2.7} = (-10)^2 \cdot (-10)^{\frac7{10}} = \underbrace{100}_{real} \cdot \underbrace{\sqrt[10]{(-10)^7}}_{imaginary}
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  8. #8
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    Hi Bjhopper
    your advise is usefull
    If i understand the exponant must be a rationnal number
    so -3^e or -3^pi can not be determined

    Thanks
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  9. #9
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    Yes, what I said about fractions was mostly wrong. I the power is irrational then the base can not be negative
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  10. #10
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    Sorry, I did not know that yout notion was valid in another country
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  11. #11
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    We can define exponentiation with irrational exponents by taking a limit.

    a^x=\lim_{n \to \infty} a^{x_n} where (x_n) is a sequence of rationals converging to x.
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  12. #12
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    Hi,
    Yes, I know what you are saying. I clearly remember my teacher showing the sequence that approached sqrt 2. I do not see (yet) why such a sequence would help us know if for example (-3)^sqrt2 is a real number.
    Thanks,
    Steven
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  13. #13
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    Dr. Steve didn't say it would help determine if such a root was real or not- only that it could be used to find the root if it existed. If \{r_n= \frac{a_n}{b_n}\} is a sequence of rational numbers converging to \sqrt{2} then for any positive real number, X, (X)^{\sqrt{2}} is defined as the limit of the sequence X^{r_n}= \sqrt[b_n]{X^{a_n}}. In general, negative real numbers to an irrational power are not defined.
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  14. #14
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    Ok, that cleared everything up for me. Thanks!
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