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Solve the exponential equation for x. {:[2^(6x-8)*64^(2-x)=2^(9x+4)],[x=]:}

Solve the exponential equation for x x .\newline26x−8⋅642−x=29x+4x=□ \begin{array}{l} 2^{6 x-8} \cdot 64^{2-x}=2^{9 x+4} \\ x=\square \end{array}

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Q. Solve the exponential equation for x x .\newline26x−8⋅642−x=29x+4x=□ \begin{array}{l} 2^{6 x-8} \cdot 64^{2-x}=2^{9 x+4} \\ x=\square \end{array}
  1. Recognize power of 22: First, we need to recognize that 6464 is a power of 22, specifically 64=2664 = 2^6. We can use this to rewrite the equation in terms of the base 22.
  2. Rewrite equation in base 22: Rewrite 642−x64^{2-x} as (26)2−x(2^6)^{2-x} and simplify the equation.\newline642−x=(26)2−x=26∗(2−x)=212−6x64^{2-x} = (2^6)^{2-x} = 2^{6*(2-x)} = 2^{12 - 6x}\newlineNow the equation is 26x−8⋅212−6x=29x+42^{6x-8} \cdot 2^{12 - 6x} = 2^{9x+4}.
  3. Combine left side of equation: Combine the left side of the equation using the property of exponents that states am×an=am+na^{m} \times a^{n} = a^{m+n}. \newline26x−8×212−6x=2(6x−8)+(12−6x)=242^{6x-8} \times 2^{12 - 6x} = 2^{(6x-8) + (12 - 6x)} = 2^{4}. \newlineNow we have 24=29x+42^{4} = 2^{9x+4}.
  4. Set exponents equal: Since the bases are the same and the equation is an equality, we can set the exponents equal to each other. 4=9x+44 = 9x + 4
  5. Subtract 44 from both sides: Subtract 44 from both sides to solve for xx.4−4=9x+4−44 - 4 = 9x + 4 - 40=9x0 = 9x
  6. Divide both sides by 99: Divide both sides by 99 to isolate xx. \newline09=9x9\frac{0}{9} = \frac{9x}{9}\newlinex=0x = 0

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