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Solve the exponential equation for x. {:[6^(4x-9)=((1)/(36))^(x-4)],[x=◻]:}

Solve the exponential equation for x x .\newline64x−9=(136)x−4x=□ \begin{array}{l} 6^{4 x-9}=\left(\frac{1}{36}\right)^{x-4} \\ x=\square \end{array}

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Full solution

Q. Solve the exponential equation for x x .\newline64x−9=(136)x−4x=□ \begin{array}{l} 6^{4 x-9}=\left(\frac{1}{36}\right)^{x-4} \\ x=\square \end{array}
  1. Write Equation: Write down the given exponential equation.\newline64x−9=(136)x−46^{4x-9} = \left(\frac{1}{36}\right)^{x-4}
  2. Use Same Base: Recognize that 136\frac{1}{36} can be written as 6−26^{-2} to have the same base on both sides of the equation.\newline64x−9=(6−2)x−46^{4x-9} = (6^{-2})^{x-4}
  3. Apply Power Rule: Apply the power of a power rule to the right side of the equation, which states that (am)n=a(m∗n)(a^m)^n = a^{(m*n)}.\newline6(4x−9)=6(−2∗(x−4))6^{(4x-9)} = 6^{(-2*(x-4))}
  4. Simplify Exponents: Simplify the exponent on the right side of the equation. 64x−9=6−2x+86^{4x-9} = 6^{-2x+8}
  5. Set Exponents Equal: Since the bases are the same and the equation is an equality, the exponents must be equal. 4x−9=−2x+84x - 9 = -2x + 8
  6. Combine Like Terms: Add 2x2x to both sides to get all xx terms on one side.\newline4x+2x−9=84x + 2x - 9 = 8\newline6x−9=86x - 9 = 8
  7. Isolate X Term: Add 99 to both sides to isolate the term with xx.\newline6x=8+96x = 8 + 9\newline6x=176x = 17
  8. Solve for X: Divide both sides by 66 to solve for xx.x=176x = \frac{17}{6}

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