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Solve the exponential equation: 2^(2x-3) imes 2^(x+2) = 128.

Mathematics

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Answer

83\frac{8}{3}

Step-by-step solution

5 steps

  1. Rewrite the equation.

    The given equation is 2(2x−3)×2(x+2)=1282^{(2x-3)} \times 2^{(x+2)} = 128.

  2. Apply the exponent rule am×an=am+na^m \times a^n = a^{m+n} to the left side of the equation.

    2(2x−3)+(x+2)=1282^{(2x-3) + (x+2)} = 128

    Combine the exponents:

    2(3x−1)=1282^{(3x-1)} = 128

  3. Express the right side of the equation as a power of 2.

    We know that 27=1282^7 = 128.
    Substitute this into the equation:

    2(3x−1)=272^{(3x-1)} = 2^7

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

    3x−1=73x - 1 = 7

  5. Solve the linear equation for xx.

    Add 1 to both sides:

    3x=7+13x = 7 + 1 3x=83x = 8

    Divide by 3:

    x=83x = \frac{8}{3}

    The value of xx is 83\boxed{\frac{8}{3}}.

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