(3)/(5)x^(-2)/(5)times (4-x)-x^(3)/(5)=0

To solve the equation $\frac{3}{5}x^{-\frac{2}{5}} \times (4 - x) - x^{\frac{3}{5}} = 0$, we can start by simplifying the equation and finding the values of $x$ that satisfy it.<br /><br />First, let's rewrite the equation in a more manageable form:<br /><br />$\frac{3}{5}x^{-\frac{2}{5}} \times (4 - x) - x^{\frac{3}{5}} = 0$<br /><br />We can factor out $x^{-\frac{2}{5}}$ from the first term:<br /><br />$x^{-\frac{2}{5}} \left(\frac{3}{5} \times (4 - x)\right) - x^{\frac{3}{5}} = 0$<br /><br />Now, let's simplify the expression inside the parentheses:<br /><br />$\frac{3}{5} \times (4 - x) = \frac{12}{5} - \frac{3}{5}x$<br /><br />So the equation becomes:<br /><br />$x^{-\frac{2}{5}} \left(\frac{12}{5} - \frac{3}{5}x\right) - x^{\frac{3}{5}} = 0$<br /><br />Next, let's distribute $x^{-\frac{2}{5}}$:<br /><br />$\frac{12}{5}x^{-\frac{2}{5}} - \frac{3}{5}x^{-\frac{2}{5} + \frac{3}{5}} - x^{\frac{3}{5}} = 0$<br /><br />Simplify the exponent in the second term:<br /><br />$x^{-\frac{2}{5} + \frac{3}{5}} = x^{\frac{1}{5}}$<br /><br />So the equation now is:<br /><br />$\frac{12}{5}x^{-\frac{2}{5}} - \frac{3}{5}x^{\frac{1}{5}} - x^{\frac{3}{5}} = 0$<br /><br />To solve this equation, we can look for values of $x$ that make each term equal to zero. Since $x^{\frac{3}{5}}$ is never zero, we focus on the other terms:<br /><br />$\frac{12}{5}x^{-\frac{2}{5}} - \frac{3}{5}x^{\frac{1}{5}} = 0$<br /><br />Factor out $x^{-\frac{2}{5}}$:<br /><br />$x^{-\frac{2}{5}} \left(\frac{12}{5} - \frac{3}{5}x\right) = 0$<br /><br />For this product to be zero, either $x^{-\frac{2}{5}} = 0$ or $\frac{12}{5} - \frac{3}{5}x = 0$.<br /><br />Since $x^{-\frac{2}{5}} = 0$ has no solution (as $x$ cannot be zero), we solve the second equation:<br /><br />$\frac{12}{5} - \frac{3}{5}x = 0$<br /><br />Multiply through by 5 to clear the fraction:<br /><br />$12 - 3x = 0$<br /><br />Solve for $x$:<br /><br />$3x = 12$<br /><br />$x = 4$<br /><br />So the solution to the equation is $x = 4$.