Mastering Exponents: Simplifying and Applying Rules for Derivative Homework

[Torshi]

Homework Statement

([3√(x^2+4)^4]^2

Homework Equations

None needed.
Chain rule
product rule etc

The Attempt at a Solution

I stopped at:

[((x^2+4)^4)^1/3]^2

So I have 3 exponents. I don't know how to simplify this in order to move on to do the chain rule or whatever rule that comes next. The exponents are killing me

 

[Mark44]

Homework Statement

([3√(x^2+4)^4]^2

Although you can't tell from the above, from your work below, it appears that the radical is a cube root.

Is this what you're trying to differentiate?
$(\sqrt[3]{(x^2 + 4)^4})^2$

Homework Equations

None needed.
Chain rule
product rule etc

The Attempt at a Solution

I stopped at:

[((x^2+4)^4)^1/3]^2

So I have 3 exponents. I don't know how to simplify this in order to move on to do the chain rule or whatever rule that comes next. The exponents are killing me

What does (a^r)^s simplify to?

 

[Torshi]

Although you can't tell from the above, from your work below, it appears that the radical is a cube root.

Is this what you're trying to differentiate?
$(\sqrt[3]{(x^2 + 4)^4})^2$What does (a^r)^s simplify to?

I think it simplified down to (x^2+4)^8/3
I multiplied the exponents: 1/3 * 4/1 * 2/1 = 8/3

 

[Mark44]

OK, that's the first step.

Now, what is d/dx[(x² + 4)^8/3]?

 

[Torshi]

OK, that's the first step.

Now, what is d/dx[(x² + 4)^8/3]?

I figured it out. Thank you. My main issue was with the exponents in regards to if I had to multiply all of them together which was true.