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Solve the exponential equation for x. {:[((1)/(64))^(-8x+3)=((1)/(16))^(7x-2)],[x=◻]:}

Solve the exponential equation for x x .\newline(164)−8x+3=(116)7x−2x=□ \begin{array}{l} \left(\frac{1}{64}\right)^{-8 x+3}=\left(\frac{1}{16}\right)^{7 x-2} \\ x=\square \end{array}

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Full solution

Q. Solve the exponential equation for x x .\newline(164)−8x+3=(116)7x−2x=□ \begin{array}{l} \left(\frac{1}{64}\right)^{-8 x+3}=\left(\frac{1}{16}\right)^{7 x-2} \\ x=\square \end{array}
  1. Rewrite bases to match: First, let's rewrite the bases of the exponents to have the same base since 6464 and 1616 are both powers of 22. We know that 64=2664 = 2^6 and 16=2416 = 2^4. So we can rewrite the equation as follows:\newline(164)−8x+3=(116)7x−2\left(\frac{1}{64}\right)^{-8x+3} = \left(\frac{1}{16}\right)^{7x-2}\newline(126)−8x+3=(124)7x−2\left(\frac{1}{2^6}\right)^{-8x+3} = \left(\frac{1}{2^4}\right)^{7x-2}
  2. Apply power of a power: Next, we apply the power of a power rule, which states that (am)n=amn(a^m)^n = a^{mn}, to both sides of the equation:\newline(2−6)−8x+3=(2−4)7x−2\left(2^{-6}\right)^{-8x+3} = \left(2^{-4}\right)^{7x-2}\newline2−6(−8x+3)=2−4(7x−2)2^{-6(-8x+3)} = 2^{-4(7x-2)}
  3. Simplify the exponents: Now we simplify the exponents on both sides:\newline248x−18=228x−82^{48x-18} = 2^{28x-8}
  4. Set exponents equal: Since the bases are the same and the equation is an equality, we can set the exponents equal to each other:\newline48x−18=28x−848x - 18 = 28x - 8
  5. Subtract and isolate x: Next, we solve for x by first subtracting 2828x from both sides:\newline48x−28x−18=28x−28x−848x - 28x - 18 = 28x - 28x - 8\newline20x−18=−820x - 18 = -8
  6. Add and isolate x: Then we add 1818 to both sides to isolate the term with x:\newline20x−18+18=−8+1820x - 18 + 18 = -8 + 18\newline20x=1020x = 10
  7. Divide to find x: Finally, we divide both sides by 2020 to solve for x:\newlinex=1020x = \frac{10}{20}\newlinex=12x = \frac{1}{2}

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