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Solve the exponential equation for x. {:[(81^(x+5))/(3^(7x+2))=3^(x-2)],[x=◻]:}

Solve the exponential equation for x x .\newline81x+537x+2=3x−2x=□ \begin{array}{l} \frac{81^{x+5}}{3^{7 x+2}}=3^{x-2} \\ x=\square \end{array}

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Full solution

Q. Solve the exponential equation for x x .\newline81x+537x+2=3x−2x=□ \begin{array}{l} \frac{81^{x+5}}{3^{7 x+2}}=3^{x-2} \\ x=\square \end{array}
  1. Expressing 8181 as a power of 33: First, we need to express 8181 as a power of 33 because 8181 is 343^4. This will allow us to have a common base for all terms in the equation.\newline(81(x+5)3(7x+2))=3(x−2) \left(\frac{81^{(x+5)}}{3^{(7x+2)}}\right) = 3^{(x-2)} \newline((34)(x+5)3(7x+2))=3(x−2) \left(\frac{(3^4)^{(x+5)}}{3^{(7x+2)}}\right) = 3^{(x-2)}
  2. Applying the power of a power rule: Next, we apply the power of a power rule, which states that (am)n=a(m∗n)(a^{m})^{n} = a^{(m*n)}, to the numerator.\newline(3(4∗(x+5))3(7x+2))=3(x−2) \left(\frac{3^{(4*(x+5))}}{3^{(7x+2)}}\right) = 3^{(x-2)} \newline(3(4x+20)3(7x+2))=3(x−2) \left(\frac{3^{(4x+20)}}{3^{(7x+2)}}\right) = 3^{(x-2)}
  3. Using the quotient of powers rule: Now, we use the quotient of powers rule, which states that aman=a(m−n)\frac{a^m}{a^n} = a^{(m-n)}, to simplify the left side of the equation.\newline3(4x+20−(7x+2))=3(x−2) 3^{(4x+20-(7x+2))} = 3^{(x-2)} \newline3(−3x+18)=3(x−2) 3^{(-3x+18)} = 3^{(x-2)}
  4. Setting the exponents equal to each other: Since the bases are the same and the equation is an equality, we can set the exponents equal to each other.\newline−3x+18=x−2 -3x + 18 = x - 2
  5. Solving for x: Now, we solve for x by moving all terms involving x to one side and constants to the other side.\newline−3x−x=−2−18 -3x - x = -2 - 18 \newline−4x=−20 -4x = -20
  6. Final solution: Finally, we divide both sides by −4-4 to solve for x.\newlinex=−20−4 x = \frac{-20}{-4} \newlinex=5 x = 5

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