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Solve the exponential equation for x. {:[2^(4-3x)=16^((2)/(5))],[x=]:}

Solve the exponential equation for x x .\newline24−3x=1625x=□ \begin{array}{l} 2^{4-3 x}=16^{\frac{2}{5}} \\ x=\square \end{array}

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Full solution

Q. Solve the exponential equation for x x .\newline24−3x=1625x=□ \begin{array}{l} 2^{4-3 x}=16^{\frac{2}{5}} \\ x=\square \end{array}
  1. Recognize Power of 22: Recognize that 1616 is a power of 22, since 16=2416 = 2^4. We can rewrite the equation using this fact to have the same base on both sides.\newline24−3x=(24)252^{4-3x} = (2^4)^{\frac{2}{5}}
  2. Apply Power Rule: Apply the power of a power rule, which states that a^b)^c = a^{(b*c)}\. \$(2^{(4-3x)}) = 2^{(4*(2/5))}
  3. Set Exponents Equal: Since the bases are now the same, we can set the exponents equal to each other. 4−3x=4(25)4 - 3x = 4\left(\frac{2}{5}\right)
  4. Simplify Right Side: Simplify the right side of the equation. 4−3x=854 - 3x = \frac{8}{5}
  5. Isolate Variable: Solve for xx by isolating the variable on one side.\newline3x=4−853x = 4 - \frac{8}{5}
  6. Convert to Fraction: Convert 44 to a fraction with a denominator of 55 to combine the terms.\newline3x=205−853x = \frac{20}{5} - \frac{8}{5}
  7. Subtract Fractions: Subtract the fractions. 3x=(205)−(85)=1253x = \left(\frac{20}{5}\right) - \left(\frac{8}{5}\right) = \frac{12}{5}
  8. Divide by 33: Divide both sides by 33 to solve for xx.x=1253x = \frac{\frac{12}{5}}{3}
  9. Simplify Division: Simplify the division by 33.x=(125)⋅(13)x = \left(\frac{12}{5}\right) \cdot \left(\frac{1}{3}\right)
  10. Multiply Numerators: Multiply the numerators and denominators. x=1215x = \frac{12}{15}
  11. Simplify Fraction: Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 33. \newlinex=45x = \frac{4}{5}

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