Why $(x^5+x)^{\frac{1}{3}}=x^\frac{5}{3}(1+\frac{1}{x^4})^{\frac{1}{3}}$?

Question

My teacher today wrote the following equation: $(x^5+x)^{\frac{1}{3}}=x^\frac{5}{3}(1+\frac{1}{x^4})^{\frac{1}{3}}$. Why is this true?

Answer 1

It is $$(x^5+x)^{1/3}=(x^5(1+\frac{1}{x^4}))^{1/3}=x^{5/3}(1+\frac{1}{x^4})^{1/3}$$

Answer 2

$$(x^5+x)^{\frac13}=\left(x^5\left(1+\frac1{x^4}\right)\right)^{\frac13}=(x^5)^{\frac13}\left(1+\frac1{x^4}\right)^{\frac13}=x^{\frac53}\left(1+\frac1{x^4}\right)^{\frac13}$$

Answer 3

hint

$$x^5+x=x^5(1+\frac{1}{x^4})$$

Answer 4

We are using that

$$(A\cdot B)^n=A^n\cdot B^n$$

and therefore we have

$$(x^5+x)^{\frac{1}{3}}=\left(x^5\cdot 1+x^5\cdot \frac{1}{x^4}\right)^{\frac{1}{3}}=(x^5)^\frac{1}{3}\left(1+\frac{1}{x^4}\right)^{\frac{1}{3}}=x^\frac{5}{3}\left(1+\frac{1}{x^4}\right)^{\frac{1}{3}}$$