Exercise
$x\left(1\right)=\frac{4x^{\frac{1}{12}}-x^{\frac{1}{3}}-3}{12x^{\frac{3}{4}}\left(x^{\frac{1}{3}}-1\right)^2}$
Step-by-step Solution
Intermediate steps
1
Expand $\left(\sqrt[3]{x}-1\right)^2$
$x\left(1\right)=\frac{4\sqrt[12]{x}-\sqrt[3]{x}-3}{12\sqrt[4]{x^{3}}\left(\sqrt[3]{x^{2}}-2\sqrt[3]{x}+1\right)}$
2
Multiply the single term $12\sqrt[4]{x^{3}}$ by each term of the polynomial $\left(\sqrt[3]{x^{2}}-2\sqrt[3]{x}+1\right)$
$\frac{4\sqrt[12]{x}-\sqrt[3]{x}-3}{12\sqrt[3]{x^{2}}\sqrt[4]{x^{3}}-2\cdot 12\sqrt[3]{x}\sqrt[4]{x^{3}}+12\sqrt[4]{x^{3}}}$
3
Multiply $-2$ times $12$
$x\left(1\right)=\frac{4\sqrt[12]{x}-\sqrt[3]{x}-3}{12\sqrt[3]{x^{2}}\sqrt[4]{x^{3}}-24\sqrt[3]{x}\sqrt[4]{x^{3}}+12\sqrt[4]{x^{3}}}$
Intermediate steps
4
When multiplying exponents with same base we can add the exponents
$x\left(1\right)=\frac{4\sqrt[12]{x}-\sqrt[3]{x}-3}{12\sqrt[12]{x^{17}}-24\sqrt[3]{x}\sqrt[4]{x^{3}}+12\sqrt[4]{x^{3}}}$
Intermediate steps
5
When multiplying exponents with same base we can add the exponents
$x\left(1\right)=\frac{4\sqrt[12]{x}-\sqrt[3]{x}-3}{12\sqrt[12]{x^{17}}-24\sqrt[12]{x^{13}}+12\sqrt[4]{x^{3}}}$
Final answer to the exercise
$x\left(1\right)=\frac{4\sqrt[12]{x}-\sqrt[3]{x}-3}{12\sqrt[12]{x^{17}}-24\sqrt[12]{x^{13}}+12\sqrt[4]{x^{3}}}$