$$ \frac { \frac { ( 5 x ^ { 2 } + 1 ) ^ { 1 / 3 } } { 3 x ^ { 1 / 3 } } - \frac { 10 x ^ { 4 / 3 } } { 3 ( 5 x ^ { 5 } + 1 ) ^ { 2 / 3 } } } { ( 5 x ^ { 2 } + 1 ) ^ { 2 / 3 } } $$

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Vamos a simplificar la expresión:

$$
\frac { \frac { ( 5 x ^ { 2 } + 1 ) ^ { 1 / 3 } } { 3 x ^ { 1 / 3 } } - \frac { 10 x ^ { 4 / 3 } } { 3 ( 5 x ^ { 5 } + 1 ) ^ { 2 / 3 } } } { ( 5 x ^ { 2 } + 1 ) ^ { 2 / 3 } }
$$

Primero, simplificaremos el numerador y luego dividiremos por el denominador.

1. **Numerador**: 
   - La primera parte es $$ \frac { ( 5 x ^ { 2 } + 1 ) ^ { 1 / 3 } } { 3 x ^ { 1 / 3 } } $$.
   - La segunda parte es $$ \frac { 10 x ^ { 4 / 3 } } { 3 ( 5 x ^ { 5 } + 1 ) ^ { 2 / 3 } } $$.

2. **Denominador**: 
   - Es $$ ( 5 x ^ { 2 } + 1 ) ^ { 2 / 3 } $$.

Ahora, vamos a realizar los cálculos necesarios para simplificar la expresión.
Simplify the expression by following steps:
- step0: Solution:
  $$\left(5x^{2}+1\right)^{\frac{2}{3}}$$
- step1: Simplify:
  $$\sqrt[3]{\left(5x^{2}+1\right)^{2}}$$
Calculate or simplify the expression $$ (5*x^2 + 1)^(1/3)/(3*x^(1/3)) - 10*x^(4/3)/(3*(5*x^5 + 1)^(2/3)) $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\frac{\left(5x^{2}+1\right)^{\frac{1}{3}}}{\left(3x^{\frac{1}{3}}\right)}-\frac{10x^{\frac{4}{3}}}{\left(3\left(5x^{5}+1\right)^{\frac{2}{3}}\right)}$$
- step1: Rewrite the expression:
  $$\frac{\left(5x^{2}+1\right)^{\frac{1}{3}}}{3x^{\frac{1}{3}}}-\frac{10x^{\frac{4}{3}}}{\left(3\left(5x^{5}+1\right)^{\frac{2}{3}}\right)}$$
- step2: Remove the parentheses:
  $$\frac{\left(5x^{2}+1\right)^{\frac{1}{3}}}{3x^{\frac{1}{3}}}-\frac{10x^{\frac{4}{3}}}{3\left(5x^{5}+1\right)^{\frac{2}{3}}}$$
- step3: Reduce fractions to a common denominator:
  $$\frac{\left(5x^{2}+1\right)^{\frac{1}{3}}\left(5x^{5}+1\right)^{\frac{2}{3}}}{3x^{\frac{1}{3}}\left(5x^{5}+1\right)^{\frac{2}{3}}}-\frac{10x^{\frac{4}{3}}\times x^{\frac{1}{3}}}{3\left(5x^{5}+1\right)^{\frac{2}{3}}x^{\frac{1}{3}}}$$
- step4: Rewrite the expression:
  $$\frac{\left(5x^{2}+1\right)^{\frac{1}{3}}\left(5x^{5}+1\right)^{\frac{2}{3}}}{3x^{\frac{1}{3}}\left(5x^{5}+1\right)^{\frac{2}{3}}}-\frac{10x^{\frac{4}{3}}\times x^{\frac{1}{3}}}{3x^{\frac{1}{3}}\left(5x^{5}+1\right)^{\frac{2}{3}}}$$
- step5: Transform the expression:
  $$\frac{\left(5x^{2}+1\right)^{\frac{1}{3}}\left(5x^{5}+1\right)^{\frac{2}{3}}-10x^{\frac{4}{3}}\times x^{\frac{1}{3}}}{3x^{\frac{1}{3}}\left(5x^{5}+1\right)^{\frac{2}{3}}}$$
- step6: Multiply the terms:
  $$\frac{\left(5x^{2}+1\right)^{\frac{1}{3}}\left(5x^{5}+1\right)^{\frac{2}{3}}-10x^{\frac{5}{3}}}{3x^{\frac{1}{3}}\left(5x^{5}+1\right)^{\frac{2}{3}}}$$
- step7: Simplify:
  $$\frac{\sqrt[3]{5x^{2}+1}\times \sqrt[3]{\left(5x^{5}+1\right)^{2}}-10x\sqrt[3]{x^{2}}}{3x^{\frac{1}{3}}\left(5x^{5}+1\right)^{\frac{2}{3}}}$$
- step8: Simplify:
  $$\frac{\sqrt[3]{5x^{2}+1}\times \sqrt[3]{\left(5x^{5}+1\right)^{2}}-10x\sqrt[3]{x^{2}}}{3\sqrt[3]{x}\times \sqrt[3]{\left(5x^{5}+1\right)^{2}}}$$
- step9: Calculate:
  $$\frac{\sqrt[3]{\left(5x^{6}+x\right)^{2}\left(5x^{2}+1\right)\left(5x^{5}+1\right)}-10x^{2}\sqrt[3]{5x^{6}+x}}{15x^{6}+3x}$$
- step10: Expand the expression:
  $$\frac{5x^{5}\sqrt[3]{5x^{4}+x^{2}}+\sqrt[3]{5x^{4}+x^{2}}-10x^{2}\sqrt[3]{5x^{6}+x}}{15x^{6}+3x}$$
Después de realizar los cálculos, tenemos:

1. **Numerador simplificado**:
   $$
   \frac{5x^{5}\sqrt[3]{5x^{4}+x^{2}}+\sqrt[3]{5x^{4}+x^{2}}-10x^{2}\sqrt[3]{5x^{6}+x}}{15x^{6}+3x}
   $$

2. **Denominador simplificado**:
   $$
   \sqrt[3]{(5x^{2}+1)^{2}}
   $$

Ahora, podemos escribir la expresión completa como:

$$
\frac{\frac{5x^{5}\sqrt[3]{5x^{4}+x^{2}}+\sqrt[3]{5x^{4}+x^{2}}-10x^{2}\sqrt[3]{5x^{6}+x}}{15x^{6}+3x}}{\sqrt[3]{(5x^{2}+1)^{2}}}
$$

Si necesitas más simplificaciones o un resultado específico, házmelo saber.
La expresión simplificada es: $$ \frac{5x^{5}\sqrt[3]{5x^{4}+x^{2}}+\sqrt[3]{5x^{4}+x^{2}}-10x^{2}\sqrt[3]{5x^{6}+x}}{15x^{6}+3x} \div \sqrt[3]{(5x^{2}+1)^{2}} $$ O, de manera más compacta: $$ \frac{5x^{5}\sqrt[3]{5x^{4}+x^{2}}+\sqrt[3]{5x^{4}+x^{2}}-10x^{2}\sqrt[3]{5x^{6}+x}}{15x^{6}+3x \cdot \sqrt[3]{(5x^{2}+1)^{2}}} $$

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