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Solve the exponential equation for x. {:[(49^(4x-1))/(7^(x-6))=7^(8x+3)],[x=◻]:}

Solve the exponential equation for x x .\newline494x−17x−6=78x+3x=□ \begin{array}{l} \frac{49^{4 x-1}}{7^{x-6}}=7^{8 x+3} \\ x=\square \end{array}

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Q. Solve the exponential equation for x x .\newline494x−17x−6=78x+3x=□ \begin{array}{l} \frac{49^{4 x-1}}{7^{x-6}}=7^{8 x+3} \\ x=\square \end{array}
  1. Simplify the equation: Simplify the equation using the properties of exponents.\newlineGiven the equation:\newline(494x−1)/(7x−6)=78x+3(49^{4x-1})/(7^{x-6}) = 7^{8x+3}\newlineWe know that 4949 is 727^2, so we can rewrite 494x−149^{4x-1} as (72)4x−1(7^2)^{4x-1}.\newlineUsing the power of a power property, (am)n=amn(a^m)^n = a^{mn}, we get:\newline(72(4x−1))/(7x−6)=78x+3(7^{2(4x-1)})/(7^{x-6}) = 7^{8x+3}\newlineSimplify the exponent in the numerator:\newline78x−2/7x−6=78x+37^{8x-2}/7^{x-6} = 7^{8x+3}
  2. Rewrite using properties of exponents: Combine the terms with the same base on the left side of the equation.\newlineUsing the quotient of powers property, am/an=am−na^m/a^n = a^{m-n}, we combine the terms:\newline7(8x−2)−(x−6)=78x+37^{(8x-2)-(x-6)} = 7^{8x+3}\newlineSimplify the exponent on the left side:\newline77x+4=78x+37^{7x+4} = 7^{8x+3}
  3. Combine terms with the same base: Since the bases are the same, we can set the exponents equal to each other.\newline7x+4=8x+37x+4 = 8x+3
  4. Set exponents equal to each other: Solve for x.\newlineSubtract 7x7x from both sides:\newline4=x+34 = x + 3\newlineSubtract 33 from both sides:\newlinex=1x = 1

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