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Solve the equation: (27^x * 9^(2x)) / (1/3)^x = 81

Mathematics

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Answer

12\text{}\frac{1}{2}\text{}

Step-by-step solution

6 steps

  1. Express all numbers in the equation as powers of 3.

    The given equation is:

    27x×92x(13)x=81\frac{27^x \times 9^{2x}}{\left(\frac{1}{3}\right)^x} = 81

    We can rewrite the bases as powers of 3:
    27=3327 = 3^3
    9=329 = 3^2
    13=3−1\frac{1}{3} = 3^{-1}
    81=3481 = 3^4

    Substitute these into the equation:

    (33)x×(32)2x(3−1)x=34\frac{(3^3)^x \times (3^2)^{2x}}{(3^{-1})^x} = 3^4

  2. Simplify the exponents using the rule (am)n=amn(a^m)^n = a^{mn}.

    33x×32×2x3−x=34\frac{3^{3x} \times 3^{2 \times 2x}}{3^{-x}} = 3^4 33x×34x3−x=34\frac{3^{3x} \times 3^{4x}}{3^{-x}} = 3^4

  3. Combine the terms in the numerator using the rule am×an=am+na^m \times a^n = a^{m+n}.

    33x+4x3−x=34\frac{3^{3x + 4x}}{3^{-x}} = 3^4 37x3−x=34\frac{3^{7x}}{3^{-x}} = 3^4

  4. Simplify the fraction using the rule aman=am−n\frac{a^m}{a^n} = a^{m-n}.

    37x−(−x)=343^{7x - (-x)} = 3^4 37x+x=343^{7x + x} = 3^4 38x=343^{8x} = 3^4

  5. Equate the exponents since the bases are the same.

    8x=48x = 4

  6. Solve for xx.

    x=48x = \frac{4}{8} x=12x = \frac{1}{2}

    The value of xx is 12\boxed{\text{}\frac{1}{2}\text{}}

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