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Solve the exponential equation for x. {:[(2^(8-x))/(32^(x+12))=2^(3x-7)],[x=◻]:}

Solve the exponential equation for x x .\newline28−x32x+12=23x−7x=□ \begin{array}{l} \frac{2^{8-x}}{32^{x+12}}=2^{3 x-7} \\ x=\square \end{array}

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Full solution

Q. Solve the exponential equation for x x .\newline28−x32x+12=23x−7x=□ \begin{array}{l} \frac{2^{8-x}}{32^{x+12}}=2^{3 x-7} \\ x=\square \end{array}
  1. Simplify the equation: First, let's simplify the equation by recognizing that 3232 is a power of 22, specifically 32=2532 = 2^5. We can then rewrite the equation using this fact.\newline(28−x32x+12)=23x−7\left(\frac{2^{8-x}}{32^{x+12}}\right) = 2^{3x-7}\newline(28−x(25)x+12)=23x−7\left(\frac{2^{8-x}}{(2^5)^{x+12}}\right) = 2^{3x-7}\newline(28−x25(x+12))=23x−7\left(\frac{2^{8-x}}{2^{5(x+12)}}\right) = 2^{3x-7}
  2. Combine the exponents: Now, we can use the property of exponents that states am−n=amana^{m-n} = \frac{a^m}{a^n} to combine the exponents in the numerator and the denominator.\newline28−x−5(x+12)=23x−72^{8-x-5(x+12)} = 2^{3x-7}\newline28−x−5x−60=23x−72^{8-x-5x-60} = 2^{3x-7}\newline2−5x−52=23x−72^{-5x-52} = 2^{3x-7}
  3. Equate the exponents: Since the bases are the same and the equation is set equal, we can equate the exponents.\newline−5x−52=3x−7-5x - 52 = 3x - 7
  4. Add 55x to both sides: Now, we solve for x by first adding 55x to both sides of the equation.\newline−5x+5x−52=3x+5x−7-5x + 5x - 52 = 3x + 5x - 7\newline−52=8x−7-52 = 8x - 7
  5. Add 77 to both sides: Next, we add 77 to both sides of the equation to isolate the term with x on one side.\newline−52+7=8x−7+7-52 + 7 = 8x - 7 + 7\newline−45=8x-45 = 8x
  6. Divide both sides by 88: Finally, we divide both sides by 88 to solve for x.\newline−458=8x8\frac{-45}{8} = \frac{8x}{8}\newlinex=−458x = \frac{-45}{8}

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