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Solve the exponential equation for x. {:[8^(5x+3)=((1)/(64))^(-(2)/(3))],[x=◻]:}

Solve the exponential equation for x x .\newline85x+3=(164)−23x=□ \begin{array}{l} 8^{5 x+3}=\left(\frac{1}{64}\right)^{-\frac{2}{3}} \\ x=\square \end{array}

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Q. Solve the exponential equation for x x .\newline85x+3=(164)−23x=□ \begin{array}{l} 8^{5 x+3}=\left(\frac{1}{64}\right)^{-\frac{2}{3}} \\ x=\square \end{array}
  1. Write Given Equation: Write down the given exponential equation and simplify the right side of the equation.\newlineGiven equation: 85x+3=(164)−238^{5x+3} = \left(\frac{1}{64}\right)^{-\frac{2}{3}}\newlineSimplify the right side: (164)−23=6423\left(\frac{1}{64}\right)^{-\frac{2}{3}} = 64^{\frac{2}{3}}\newlineSince 64=8264 = 8^2, we can write 642364^{\frac{2}{3}} as (82)23(8^2)^{\frac{2}{3}}.
  2. Simplify Right Side: Apply the power of a power rule to the right side of the equation.\newlineUsing the power of a power rule: (82)23=82⋅23=843(8^2)^{\frac{2}{3}} = 8^{2 \cdot \frac{2}{3}} = 8^{\frac{4}{3}}\newlineNow the equation is: 85x+3=8438^{5x+3} = 8^{\frac{4}{3}}
  3. Apply Power Rule: Since the bases are the same, set the exponents equal to each other.\newline5x+3=435x + 3 = \frac{4}{3}
  4. Set Exponents Equal: Solve for x by isolating x on one side of the equation.\newlineSubtract 33 from both sides: 5x=43−35x = \frac{4}{3} - 3\newlineConvert 33 to a fractions" target="_blank" class="backlink">fraction with a denominator of 33: 5x=43−935x = \frac{4}{3} - \frac{9}{3}\newlineCombine the fractions: 5x=4−935x = \frac{4 - 9}{3}\newlineSimplify the right side: 5x=−535x = \frac{-5}{3}
  5. Solve for x: Divide both sides by 55 to solve for x.\newlinex=−53⋅15x = \frac{-5}{3} \cdot \frac{1}{5}\newlineSimplify the right side: x=−515x = \frac{-5}{15}\newlineReduce the fraction: x=−13x = \frac{-1}{3}

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