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Solve the exponential equation for x. {:[(81^(x+7))/(9^(5x-9))=9^(4x+1)],[x=]:}

Solve the exponential equation for x x .\newline81x+795x−9=94x+1x=□ \begin{array}{l} \frac{81^{x+7}}{9^{5 x-9}}=9^{4 x+1} \\ x=\square \end{array}

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Full solution

Q. Solve the exponential equation for x x .\newline81x+795x−9=94x+1x=□ \begin{array}{l} \frac{81^{x+7}}{9^{5 x-9}}=9^{4 x+1} \\ x=\square \end{array}
  1. Recognize and Rewrite: Simplify the given exponential equation using properties of exponents.\newlineThe equation is:\newline[81(x+7)9(5x−9)=9(4x+1)][\frac{81^{(x+7)}}{9^{(5x-9)}}=9^{(4x+1)}]\newlineFirst, recognize that 8181 is a power of 99, specifically 81=9281 = 9^2. Rewrite 8181 in terms of 99:\newline[(92)(x+7)9(5x−9)=9(4x+1)][\frac{(9^2)^{(x+7)}}{9^{(5x-9)}}=9^{(4x+1)}]\newlineNow apply the power of a power rule (am)n=amn(a^m)^n = a^{mn}:\newline[92(x+7)9(5x−9)=9(4x+1)][\frac{9^{2(x+7)}}{9^{(5x-9)}}=9^{(4x+1)}]
  2. Simplify Exponent: Simplify the exponent in the numerator.\newlineUsing the power of a power rule, multiply the exponents:\newline92x+14/(95x−9)=94x+19^{2x + 14}/(9^{5x-9}) = 9^{4x+1}
  3. Set Exponents Equal: Since the bases are the same, we can set the exponents equal to each other. \newline2x+14−(5x−9)=4x+12x + 14 - (5x - 9) = 4x + 1
  4. Simplify Equation: Simplify the equation by distributing the negative sign and combining like terms.\newline2x+14−5x+9=4x+12x + 14 - 5x + 9 = 4x + 1\newline−3x+23=4x+1-3x + 23 = 4x + 1
  5. Move Terms: Move all terms involving xx to one side and constants to the other side.\newlineAdd 3x3x to both sides and subtract 11 from both sides:\newline−3x+3x+23−1=4x+3x+1−1-3x + 3x + 23 - 1 = 4x + 3x + 1 - 1\newline23−1=7x23 - 1 = 7x\newline22=7x22 = 7x
  6. Solve for x: Solve for x by dividing both sides by 77.x=227x = \frac{22}{7}

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