\( y = \frac { 1 } { \sqrt[ 3 ] { x ^ { 1 / 3 } } } + \frac { 4 x ^ { 1 / 3 } } { \sqrt[ 4 ] { x ^ { 1 / 3 } } } = \frac { \frac { 9 x ^ { 1 / 4 } } { 2 } } { \sqrt[ 5 ] { x ^ { - 1 / 3 } } } \)

To solve the equation \[ y = \frac { 1 } { \sqrt[ 3 ] { x ^ { 1 / 3 } } } + \frac { 4 x ^ { 1 / 3 } } { \sqrt[ 4 ] { x ^ { 1 / 3 } } } = \frac { \frac { 9 x ^ { 1 / 4 } } { 2 } } { \sqrt[ 5 ] { x ^ { - 1 / 3 } } }, \] we will simplify each term step by step. ### Step 1: Simplify the left-hand side 1. **First term**: \[ \frac{1}{\sqrt[3]{x^{1/3}}} = \frac{1}{x^{1/9}} = x^{-1/9}. \] 2. **Second term**: \[ \frac{4x^{1/3}}{\sqrt[4]{x^{1/3}}} = \frac{4x^{1/3}}{(x^{1/3})^{1/4}} = \frac{4x^{1/3}}{x^{1/12}} = 4x^{1/3 - 1/12} = 4x^{4/12 - 1/12} = 4x^{3/12} = 4x^{1/4}. \] Combining these two terms gives: \[ y = x^{-1/9} + 4x^{1/4}. \] ### Step 2: Simplify the right-hand side 1. **Numerator**: \[ \frac{9x^{1/4}}{2}. \] 2. **Denominator**: \[ \sqrt[5]{x^{-1/3}} = (x^{-1/3})^{1/5} = x^{-1/15}. \] Thus, the right-hand side becomes: \[ \frac{\frac{9x^{1/4}}{2}}{x^{-1/15}} = \frac{9x^{1/4}}{2} \cdot x^{1/15} = \frac{9}{2} x^{1/4 + 1/15}. \] To combine the exponents: \[ \frac{1}{4} = \frac{15}{60}, \quad \frac{1}{15} = \frac{4}{60} \quad \Rightarrow \quad \frac{1}{4} + \frac{1}{15} = \frac{15}{60} + \frac{4}{60} = \frac{19}{60}. \] So, the right-hand side simplifies to: \[ \frac{9}{2} x^{19/60}. \] ### Step 3: Set the left-hand side equal to the right-hand side Now we have: \[ x^{-1/9} + 4x^{1/4} = \frac{9}{2} x^{19/60}. \] ### Step 4: Solve for \(x\) To solve this equation, we can express all terms with a common exponent. The least common multiple of the denominators \(9\), \(4\), and \(60\) is \(180\). 1. Rewrite each term: - \(x^{-1/9} = x^{-20/180}\) - \(4x^{1/4} = 4x^{45/180}\) - \(\frac{9}{2} x^{19/60} = \frac{9}{2} x^{57/180}\) Thus, the equation becomes: \[ x^{-20/180} + 4x^{45/180} = \frac{9}{2} x^{57/180}. \] ### Step 5: Multiply through by \(x^{20/180}\) to eliminate the negative exponent This gives: \[ 1 + 4x^{65/180} = \frac{9}{2} x^{77/180}. \] ### Step 6: Rearranging the equation Rearranging gives: \[ \frac{9}{2} x^{77/180} - 4x^{65/180} - 1 = 0. \] This is a polynomial equation in terms of \(x^{1/180}\). Let \(u = x^{1/180}\), then we can rewrite the equation as: \[ \frac{9}{2} u^{77} - 4u^{65} - 1 = 0. \] ###