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(32)3⋅3x=37⋅33/311(3^2)^3 \cdot 3^x = 3^7 \cdot 3^3 / 3^{11}

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Full solution

Q. (32)3⋅3x=37⋅33/311(3^2)^3 \cdot 3^x = 3^7 \cdot 3^3 / 3^{11}
  1. Simplify power of a power: First, simplify the left side of the equation by calculating the power of a power.\newline(32)3=3(2∗3)=36(3^2)^3 = 3^{(2*3)} = 3^6
  2. Rewrite using simplified term: Now, rewrite the left side of the equation using the simplified term. 36⋅3x=36+x3^6 \cdot 3^x = 3^{6+x}
  3. Combine and divide powers: Next, simplify the right side of the equation by combining the powers of 33 that are being multiplied and then dividing.37⋅33311=37+3311=310311\frac{3^7 \cdot 3^3}{3^{11}} = \frac{3^{7+3}}{3^{11}} = \frac{3^{10}}{3^{11}}
  4. Simplify division of powers: Now, simplify the division of powers on the right side by subtracting the exponents. 310/311=3(10−11)=3−13^{10} / 3^{11} = 3^{(10-11)} = 3^{-1}
  5. Set exponents equal: We now have the simplified equation:\newline36+x=3−13^{6+x} = 3^{-1}\newlineSince the bases are the same, we can set the exponents equal to each other.\newline6+x=−16 + x = -1
  6. Solve for x: Finally, solve for x by subtracting 66 from both sides of the equation.\newlinex=−1−6x = -1 - 6\newlinex=−7x = -7

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