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Solve the exponential equation for x. {:[5^(2x+7)=25^((1)/(3))],[x=]:}

Solve the exponential equation for x x .\newline52x+7=2513x= \begin{array}{l} 5^{2 x+7}=25^{\frac{1}{3}} \\ x= \end{array}

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Full solution

Q. Solve the exponential equation for x x .\newline52x+7=2513x= \begin{array}{l} 5^{2 x+7}=25^{\frac{1}{3}} \\ x= \end{array}
  1. Rewrite as Power of 55: Rewrite 2525 as 525^2 because 2525 is a power of 55.\newline52x+7=(52)1/35^{2x+7} = (5^2)^{1/3}
  2. Apply Power Rule: Apply the power of a power rule: (ab)c=a(b∗c)(a^b)^c = a^{(b*c)}.\newline5(2x+7)=5(2∗(13))5^{(2x+7)} = 5^{(2*(\frac{1}{3}))}
  3. Set Exponents Equal: Since the bases are the same, set the exponents equal to each other. 2x+7=2×(13)2x + 7 = 2\times\left(\frac{1}{3}\right)
  4. Multiply and Subtract: Multiply 22 by 13\frac{1}{3}.\newline2x+7=232x + 7 = \frac{2}{3}
  5. Convert to Fraction: Subtract 77 from both sides to solve for xx. \newline2x=23−72x = \frac{2}{3} - 7
  6. Subtract Fractions: Convert 77 to a fraction with a denominator of 33 to combine with 23\frac{2}{3}. \newline2x=23−2132x = \frac{2}{3} - \frac{21}{3}
  7. Divide to Solve for x: Subtract the fractions. \newline2x=−1932x = -\frac{19}{3}
  8. Divide and Simplify: Divide both sides by 22 to solve for xx.x=−193/2x = \frac{-19}{3} / 2
  9. Divide and Simplify: Divide both sides by 22 to solve for xx.x=−193/2x = \frac{-19}{3} / 2Divide −19-19 by 33 and then by 22.x=−196x = \frac{-19}{6}

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