Question

question
find the positive solution of the equation.
$7x^{\frac{6}{5}} - 12 = 5091$

Explanation:

Step1: Isolate the term with \( x \)

Add 12 to both sides of the equation \( 7x^{\frac{6}{5}} - 12 = 5091 \) to get \( 7x^{\frac{6}{5}} = 5091 + 12 \).
\( 7x^{\frac{6}{5}} = 5103 \)

Step2: Solve for \( x^{\frac{6}{5}} \)

Divide both sides by 7: \( x^{\frac{6}{5}} = \frac{5103}{7} \).
\( x^{\frac{6}{5}} = 729 \)

Step3: Eliminate the fractional exponent

Raise both sides to the power of \( \frac{5}{6} \): \( x = 729^{\frac{5}{6}} \).
Since \( 729 = 3^6 \), we have \( x = (3^6)^{\frac{5}{6}} \).
Using the exponent rule \( (a^m)^n = a^{mn} \), we get \( x = 3^{6 \times \frac{5}{6}} = 3^5 \).
\( x = 243 \)

Answer:

\( 243 \)