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Solve the exponential equation for x. {:[((1)/(64))^(-2x+6)=((1)/(16))^(8x-5)],[x=◻]:}

Solve the exponential equation for x x .\newline(164)−2x+6=(116)8x−5x=□ \begin{array}{l} \left(\frac{1}{64}\right)^{-2 x+6}=\left(\frac{1}{16}\right)^{8 x-5} \\ x=\square \end{array}

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Full solution

Q. Solve the exponential equation for x x .\newline(164)−2x+6=(116)8x−5x=□ \begin{array}{l} \left(\frac{1}{64}\right)^{-2 x+6}=\left(\frac{1}{16}\right)^{8 x-5} \\ x=\square \end{array}
  1. Write Given Exponential Equation: First, let's write down the given exponential equation:\newline((1/64))(−2x+6)=((1/16))(8x−5)((1/64))^(-2x+6) = ((1/16))^(8x-5)\newlineWe need to solve for xx.
  2. Express Given Values as Powers of 22: Recognize that both sides of the equation are powers of 22. We can express 164\frac{1}{64} and 116\frac{1}{16} as powers of 22:\newline164=2−6\frac{1}{64} = 2^{-6} and 116=2−4\frac{1}{16} = 2^{-4}.\newlineRewrite the equation using these expressions:\newline(2−6)−2x+6=(2−4)8x−5(2^{-6})^{-2x+6} = (2^{-4})^{8x-5}
  3. Apply Power of a Power Rule: Apply the power of a power rule, which states that (am)n=amn(a^m)^n = a^{mn}, to both sides of the equation:\newline2(−6∗−2x+6∗−6)=2(−4∗8x−4∗−5)2^{(-6*-2x+6*-6)} = 2^{(-4*8x-4*-5)}
  4. Simplify Exponents: Simplify the exponents on both sides of the equation: 212x−36=2−32x+202^{12x-36} = 2^{-32x+20}
  5. Set Exponents Equal: Since the bases are the same and the equation is an equality, we can set the exponents equal to each other: 12x−36=−32x+2012x - 36 = -32x + 20
  6. Solve for x: Solve for x by adding 32x32x to both sides and adding 3636 to both sides:\newline12x+32x=20+3612x + 32x = 20 + 36\newline44x=5644x = 56
  7. Isolate xx: Divide both sides by 4444 to isolate xx:x=5644x = \frac{56}{44}
  8. Simplify Fraction: Simplify the fraction to find the value of xx:x=75.5x = \frac{7}{5.5}
  9. Final Value of xx: Further simplify the fraction by dividing 77 by 5.55.5:x=1.27272727272...x = 1.27272727272...

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