Find the positive solution of the equation. $\newline$ $5 x^{\frac{6}{5}}+11=2657216$ $\newline$ Answer:

Full solution

Q. Find the positive solution of the equation.

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5 x^{\frac{6}{5}}+11=2657216

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Answer:

Subtract $11$ : Subtract $11$ from both sides of the equation to isolate the term with the variable $x$ . $\newline$ $5x^{(6/5)} + 11 - 11 = 2657216 - 11$ $\newline$ $5x^{(6/5)} = 2657205$
Divide by $5$ : Divide both sides of the equation by $5$ to further isolate $x^{\frac{6}{5}}$ . $\newline$ $\frac{5x^{\frac{6}{5}}}{5} = \frac{2657205}{5}$ $\newline$ $x^{\frac{6}{5}} = 531441$
Recognize perfect power: Recognize that $531441$ is a perfect power, specifically $3^{12}$ . $\newline$ Check if $531441$ is indeed $3^{12}$ . $\newline$ $3^{12} = 3^2 \times 3^2 \times 3^2 \times 3^2 \times 3^2 \times 3^2 = 9 \times 9 \times 9 \times 9 \times 9 \times 9 = 531441$
Write $x^{\frac{6}{5}} = 3^{12}$ : Since $x^{\frac{6}{5}} = 531441$ and $531441 = 3^{12}$ , we can write $x^{\frac{6}{5}} = 3^{12}$ . Now, we need to find $x$ such that when raised to the power of $\frac{6}{5}$ , it equals $3^{12}$ .
Take $5$ th root: Take the $5$ th root of both sides of the equation to eliminate the fractional exponent. $\newline$ $(x^{\frac{6}{5}})^{\frac{5}{6}} = (3^{12})^{\frac{5}{6}}$ $\newline$ $x = 3^{12\cdot(\frac{5}{6})}$ $\newline$ $x = 3^{10}$
Calculate $x$ : Calculate $3^{10}$ to find the value of $x$ . $\newline$ $3^{10} = 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 = 59049$