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Solve the exponential equation for x. {:[2^(3x+5)=64^(x-7)],[x=◻]:}

Solve the exponential equation for x x .\newline23x+5=64x−7x=□ \begin{array}{l} 2^{3 x+5}=64^{x-7} \\ x=\square \end{array}

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Full solution

Q. Solve the exponential equation for x x .\newline23x+5=64x−7x=□ \begin{array}{l} 2^{3 x+5}=64^{x-7} \\ x=\square \end{array}
  1. Identify Power of 22: Recognize that 6464 is a power of 22, since 64=2664 = 2^6. Rewrite the equation using this fact.\newline23x+5=(26)x−72^{3x+5} = (2^6)^{x-7}
  2. Apply Power Rule: Apply the power of a power rule to simplify the right side of the equation.\newline23x+5=26(x−7)2^{3x+5} = 2^{6(x-7)}
  3. Set Exponents Equal: Since the bases are the same, set the exponents equal to each other. 3x+5=6(x−7)3x + 5 = 6(x - 7)
  4. Distribute and Simplify: Distribute the 66 on the right side of the equation.3x+5=6x−423x + 5 = 6x - 42
  5. Rearrange Terms: Move all terms involving xx to one side and constants to the other side.\newlineSubtract 3x3x from both sides:\newline5=3x−425 = 3x - 42\newlineAdd 4242 to both sides:\newline47=3x47 = 3x
  6. Solve for x: Divide both sides by 33 to solve for x.\newlinex=473x = \frac{47}{3}
  7. Calculate Value: Calculate the value of xx.x=15.666…x = 15.666\ldots

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