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Mathematics

ScanSolve AI Answer

Answer

18

Step-by-step solution

4 steps

  1. Recall the algebraic identity for the cube of a sum.

    The identity is (a+b)3=a3+b3+3ab(a+b)(a+b)^3 = a^3 + b^3 + 3ab(a+b).

  2. Apply the identity to the given expression.

    Let a=xa = x and b=1xb = \frac{1}{x}.
    Then, a+b=x+1xa+b = x + \frac{1}{x} and ab=x⋅1x=1ab = x \cdot \frac{1}{x} = 1.
    Substitute these into the identity:

    (x+1x)3=x3+(1x)3+3⋅x⋅1x(x+1x)\left(x + \frac{1}{x}\right)^3 = x^3 + \left(\frac{1}{x}\right)^3 + 3 \cdot x \cdot \frac{1}{x} \left(x + \frac{1}{x}\right) (x+1x)3=x3+1x3+3(x+1x)\left(x + \frac{1}{x}\right)^3 = x^3 + \frac{1}{x^3} + 3 \left(x + \frac{1}{x}\right)

  3. Substitute the given value into the equation.

    We are given that x+1x=3x + \frac{1}{x} = 3. Substitute this value into the equation from Step 2:

    (3)3=x3+1x3+3(3)(3)^3 = x^3 + \frac{1}{x^3} + 3(3) 27=x3+1x3+927 = x^3 + \frac{1}{x^3} + 9

  4. Solve for x3+1x3x^3 + \frac{1}{x^3}.

    Subtract 9 from both sides of the equation:

    x3+1x3=27−9x^3 + \frac{1}{x^3} = 27 - 9 x3+1x3=18x^3 + \frac{1}{x^3} = 18

    The value of x3+1x3x^3 + \frac{1}{x^3} is:

    1818
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