Question

Describe how to transform the quantity of the sixth root of x to the fifth power, to the seventh power into an expression with a rational exponent.

Answer 1

Let's solve this problem assuming the original expression is:

`(\sqrt[6]{x^5})^7`

So, we need to use some properties of exponentials:

`\begin{gathered} \sqrt[n]{y}^{}=y^{(1)/(n)} \\ \\ (y^a)^b=y^(a\cdot b) \end{gathered}`

So, let's use:

`\begin{gathered} y=x^5 \\ \\ n=6 \end{gathered}`

We have:

`\sqrt[6]{x^5}=(x^5)^{(1)/(6)}=x^{5\cdot(1)/(6)}=x^{(5)/(6)}`

Now, using the second property again, we obtain:

`(\sqrt[6]{x^5})^7=(x^{(5)/(6)})^7=x^{(5)/(6)\cdot7}=x^{(35)/(6)}`

Therefore, the expression with a rational exponent is:

`x^{(35)/(6)}`