# Mastering Exponents: Simplifying and Applying Rules for Derivative Homework

• Torshi
In summary, the problem involves differentiating a complex expression with multiple exponents, including a cube root and a power of 2. The first step is to simplify the expression by multiplying the exponents, resulting in (x^2+4)^8/3. The next step is to use the chain rule and product rule to differentiate the expression.
Torshi

([3√(x^2+4)^4]^2

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

Torshi said:

## 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##
Torshi said:

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 (ar)s simplify to?

Mark44 said:
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 (ar)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

OK, that's the first step.

Now, what is d/dx[(x2 + 4)8/3]?

Mark44 said:
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.

## 1. What are exponents and why are they important in calculus?

Exponents are mathematical notation used to represent repeated multiplication of a number by itself. They are important in calculus because they allow us to express and manipulate very large or very small numbers, and they are essential for understanding and solving equations involving variables.

## 2. What is the rule for simplifying exponents?

The rule for simplifying exponents is that when multiplying two exponents with the same base, we can add the exponents together. For example, 2^3 * 2^2 = 2^(3+2) = 2^5.

## 3. How do you apply the power rule for derivatives?

The power rule for derivatives states that when differentiating a term with an exponent, we can bring the exponent down as a coefficient and subtract 1 from the original exponent. For example, the derivative of x^3 would be 3x^(3-1) = 3x^2.

## 4. Can you simplify exponents with negative or fractional powers?

Yes, exponents can be simplified even if they are negative or fractional. For negative exponents, we can use the rule that a^-n = 1/a^n. For fractional exponents, we can use the rule that a^(m/n) = nth root of a^m.

## 5. How can mastering exponents help in solving more complex calculus problems?

Mastering exponents is crucial in solving more complex calculus problems as it allows us to simplify equations and expressions, which makes them easier to manipulate and solve. It also helps us understand the relationships between different terms and variables in a problem, making it easier to identify key patterns and concepts.

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