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Solve the exponential equation for x. {:[2^(6x+7)=((1)/(8))^(1-5x)],[x=◻]:}

Solve the exponential equation for x x .\newline26x+7=(18)1−5xx=□ \begin{array}{l} 2^{6 x+7}=\left(\frac{1}{8}\right)^{1-5 x} \\ x=\square \end{array}

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Full solution

Q. Solve the exponential equation for x x .\newline26x+7=(18)1−5xx=□ \begin{array}{l} 2^{6 x+7}=\left(\frac{1}{8}\right)^{1-5 x} \\ x=\square \end{array}
  1. Identify base of exponents: Identify the base of the exponents on both sides of the equation to see if they can be made the same.\newlineSince 23=82^3 = 8, we can rewrite 18\frac{1}{8} as 2−32^{-3}.
  2. Rewrite equation with same base: Rewrite the equation using the same base for both exponents.\newline26x+7=(2−3)1−5x2^{6x+7} = (2^{-3})^{1-5x}
  3. Apply power of a power rule: Apply the power of a power rule to the right side of the equation.\newline26x+7=2−3(1−5x)2^{6x+7} = 2^{-3(1-5x)}
  4. Set exponents equal to each other: Since the bases are now the same, we can set the exponents equal to each other.\newline6x+7=−3(1−5x)6x + 7 = -3(1 - 5x)
  5. Distribute −3-3 on right side: Distribute the −3-3 on the right side of the equation.\newline6x+7=−3+15x6x + 7 = -3 + 15x
  6. Move terms involving x: Move all terms involving x to one side and constant terms to the other side.\newline6x−15x=−3−76x - 15x = -3 - 7
  7. Combine like terms: Combine like terms.\newline−9x=−10-9x = -10
  8. Divide both sides by −9-9: Divide both sides by −9-9 to solve for x.\newlinex=−10−9x = \frac{-10}{-9}
  9. Simplify the fraction: Simplify the fraction.\newlinex=109x = \frac{10}{9}

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