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Math Help - Find the value of M

  1. #1
    Junior Member Dragon's Avatar
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    Find the value of M

    If m+n=3 and m^2+N^2=6 find the numerical value for m^3+N^3
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  2. #2
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    Quote Originally Posted by Dragon View Post
    If m+n=3 and m^2+N^2=6 find the numerical value for m^3+N^3
    You are told that,
    n+m=3
    Thus, squaring
    (n+m)^2=9
    n^2+2nm+m^2=9
    Thus,
    2mn+6=9
    Thus,
    2 mn=3 \,
    mn=1.5.

    Thus,
    n^3+m^3=(n+m)(n^2-nm+m^2)=(3)(6-1.5)
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  3. #3
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    Hello, Dragon!

    Another approach . . .


    If m+n\,=\,3 and m^2+n^2\,=\,6,
    find the numerical value for m^3+n^3

    We are given: . \begin{array}{cc}(1)\\(2)\end{array} \begin{array}{cc}m + n \:=\:3 \\ m^2+n^2\:=\:6\end{array}

    Square (1): . (m + n)^2\:=\:3^2

    . . . . . . . . . \underbrace{m^2 + n^2} + 2mn\:=\:9
    . . . . . . . . . - - \downarrow
    Substitute (2): . 6 + 2mn \:=\:9\quad\Rightarrow\quad mn \,=\,\frac{3}{2}\quad(3)


    Cube (1): . (m+n)^3\:=\:3^3\quad\Rightarrow\quad m^3 + 3m^2n + 3mn^2 + n^3\:=\:27

    . . . . . . . \text{and we have: }\;\;m^3 + n^3 + 3\underbrace{mn}\underbrace{(m + n)}\:=\:27
    . . . . . . . . . . . . . . . . . . . . . . . . . . \downarrow\quad\;\downarrow
    Substitute (3) and (1): . m^3 + n^3 + 3\left(\frac{3}{2}\right)(3)\:=\:27

    Therefore: . m^3+n^3\:=\:\frac{27}{2}

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