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Solve the exponential equation for x. {:[2^(9x+2)=16^(5x-2)],[x=]:}

Solve the exponential equation for x x .\newline29x+2=165x−2x=□ \begin{array}{l} 2^{9 x+2}=16^{5 x-2} \\ x=\square \end{array}

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Full solution

Q. Solve the exponential equation for x x .\newline29x+2=165x−2x=□ \begin{array}{l} 2^{9 x+2}=16^{5 x-2} \\ x=\square \end{array}
  1. Recognize power of 22: Recognize that 1616 is a power of 22, specifically 16=2416 = 2^4. Rewrite the equation using this fact to have the same base on both sides. 29x+2=(24)5x−22^{9x+2} = (2^4)^{5x-2}
  2. Rewrite equation with same base: Apply the power of a power rule to the right side of the equation.\newline29x+2=24⋅(5x−2)2^{9x+2} = 2^{4\cdot(5x-2)}
  3. Apply power of a power rule: Set the exponents equal to each other since the bases are now the same.\newline9x+2=4×(5x−2)9x + 2 = 4\times(5x - 2)
  4. Set exponents equal: Distribute the 44 on the right side of the equation.9x+2=20x−89x + 2 = 20x - 8
  5. Distribute on right side: Move all terms involving xx to one side of the equation and constants to the other side.9x−20x=−8−29x - 20x = -8 - 2
  6. Move terms involving xx: Combine like terms.\newline−11x=−10-11x = -10
  7. Combine like terms: Divide both sides by −11-11 to solve for xx.x=−10−11x = \frac{-10}{-11}
  8. Divide both sides: Simplify the fraction. x=1011x = \frac{10}{11}

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