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Solve the exponential equation for x. {:[4^(x-10)=((1)/(64))^(5x+2)],[x=◻]:}

Solve the exponential equation for x x .\newline4x−10=(164)5x+2x=□ \begin{array}{l} 4^{x-10}=\left(\frac{1}{64}\right)^{5 x+2} \\ x=\square \end{array}

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Q. Solve the exponential equation for x x .\newline4x−10=(164)5x+2x=□ \begin{array}{l} 4^{x-10}=\left(\frac{1}{64}\right)^{5 x+2} \\ x=\square \end{array}
  1. Write Given Exponential Equation: Write down the given exponential equation.\newline4(x−10)=(164)(5x+2)4^{(x-10)} = \left(\frac{1}{64}\right)^{(5x+2)}
  2. Rewrite Using Common Base: Recognize that 44 and 6464 are both powers of 22, so we can rewrite the equation using a common base.\newline4=224 = 2^2 and 64=2664 = 2^6, so the equation becomes:\newline(22)(x−10)=(1(26))(5x+2)(2^2)^{(x-10)} = \left(\frac{1}{(2^6)}\right)^{(5x+2)}
  3. Simplify Exponents: Simplify the exponents on both sides of the equation.\newline22(x−10)=(2−6)(5x+2)2^{2(x-10)} = (2^{-6})^{(5x+2)}\newlineThis simplifies to:\newline22x−20=2−30x−122^{2x-20} = 2^{-30x-12}
  4. Set Exponents Equal: Since the bases are the same, we can set the exponents equal to each other.\newline2x−20=−30x−122x - 20 = -30x - 12
  5. Solve for x: Solve for x by combining like terms.\newlineAdd 30x30x to both sides:\newline2x+30x−20=−30x+30x−122x + 30x - 20 = -30x + 30x - 12\newline32x−20=−1232x - 20 = -12
  6. Divide to Find x: Add 2020 to both sides to isolate the term with x.\newline32x−20+20=−12+2032x - 20 + 20 = -12 + 20\newline32x=832x = 8
  7. Divide to Find x: Add 2020 to both sides to isolate the term with x.\newline32x−20+20=−12+2032x - 20 + 20 = -12 + 20\newline32x=832x = 8Divide both sides by 3232 to solve for x.\newlinex=832x = \frac{8}{32}\newlinex=14x = \frac{1}{4}

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