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Solve the exponential equation for x. {:[32^((x)/(3))=8^(x-12)],[x=◻]:}

Solve the exponential equation for x x .\newline32x3=8x−12x=□ \begin{array}{l} 32^{\frac{x}{3}}=8^{x-12} \\ x=\square \end{array}

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Q. Solve the exponential equation for x x .\newline32x3=8x−12x=□ \begin{array}{l} 32^{\frac{x}{3}}=8^{x-12} \\ x=\square \end{array}
  1. Recognize Powers of 22: Recognize that both 3232 and 88 are powers of 22.\newline32=2532 = 2^5 and 8=238 = 2^3.\newlineRewrite the equation using this fact.\newline32x3=(25)x332^{\frac{x}{3}} = (2^5)^{\frac{x}{3}} and 8x−12=(23)x−128^{x-12} = (2^3)^{x-12}.
  2. Apply Power Rule: Apply the power of a power rule to simplify the exponents.\newline(25)x3=25x3(2^5)^{\frac{x}{3}} = 2^{\frac{5x}{3}} and (23)x−12=23(x−12)(2^3)^{x-12} = 2^{3(x-12)}.\newlineNow the equation is 25x3=23(x−12)2^{\frac{5x}{3}} = 2^{3(x-12)}.
  3. Set Exponents Equal: Since the bases are the same, set the exponents equal to each other.\newline5x3=3(x−12)\frac{5x}{3} = 3(x-12).
  4. Solve for x: Solve for x by first distributing the 33 on the right side of the equation.\newline5x3=3x−36\frac{5x}{3} = 3x - 36.
  5. Clear Fraction: Multiply every term by 33 to clear the fraction.\newline5x=9x−1085x = 9x - 108.
  6. Move Terms: Move all x terms to one side and constants to the other side.\newline5x−9x=−1085x - 9x = -108.
  7. Combine Like Terms: Combine like terms.\newline−4x=−108-4x = -108.
  8. Divide by −4-4: Divide both sides by −4-4 to solve for x.\newlinex=−108−4x = \frac{-108}{-4}.
  9. Calculate x: Calculate the value of x.\newlinex=27x = 27.

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