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Solve the exponential equation for x. {:[9^(3x-10)=((1)/(81))^((1)/(6))],[x=◻]:}

Solve the exponential equation for x x .\newline93x−10=(181)16x=□ \begin{array}{l} 9^{3 x-10}=\left(\frac{1}{81}\right)^{\frac{1}{6}} \\ x=\square \end{array}

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Q. Solve the exponential equation for x x .\newline93x−10=(181)16x=□ \begin{array}{l} 9^{3 x-10}=\left(\frac{1}{81}\right)^{\frac{1}{6}} \\ x=\square \end{array}
  1. Identify Base: Identify the base of the exponential expressions on both sides of the equation.\newline93x−109^{3x-10} has a base of 99, and (181)16\left(\frac{1}{81}\right)^{\frac{1}{6}} can be rewritten with a base of 99 because 81=9281 = 9^2.
  2. Rewrite Right Side: Rewrite the right side of the equation using the base of 99.\newline(181)16=(192)16=(9−2)16=9−26=9−13\left(\frac{1}{81}\right)^{\frac{1}{6}} = \left(\frac{1}{9^2}\right)^{\frac{1}{6}} = (9^{-2})^{\frac{1}{6}} = 9^{-\frac{2}{6}} = 9^{-\frac{1}{3}}.
  3. Set Exponents Equal: Set the exponents equal to each other since the bases are now the same.\newline3x−10=−133x - 10 = -\frac{1}{3}.
  4. Solve for x: Solve for x by isolating the variable.\newlineAdd 1010 to both sides: 3x=−13+103x = -\frac{1}{3} + 10.
  5. Convert to Fraction: Convert 1010 to a fraction with a denominator of 33 to combine with −13-\frac{1}{3}.\newline10=10⋅33=30310 = \frac{10 \cdot 3}{3} = \frac{30}{3}.
  6. Combine Fractions: Combine the fractions.\newline3x=−13+303=−1+303=2933x = -\frac{1}{3} + \frac{30}{3} = \frac{-1 + 30}{3} = \frac{29}{3}.
  7. Divide by 33: Divide both sides by 33 to solve for x.\newlinex=293⋅13=299x = \frac{29}{3} \cdot \frac{1}{3} = \frac{29}{9}.

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