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The given expression to simplify is:

\[
\frac{\frac{c + \frac{64}{c^2}}{1 + \frac{4}{c}}}{}
\]

Let's go step by step to simplify this.

### Step 1: Simplify the numerator
The numerator is:
\[
c + \frac{64}{c^2}
\]
To combine these, you need to get a common denominator:
\[
\frac{c}{1} + \frac{64}{c^2} = \frac{c \cdot c^2 + 64}{c^2} = \frac{c^3 + 64}{c^2}
\]

### Step 2: Simplify the denominator
The denominator is:
\[
1 + \frac{4}{c}
\]
To combine these, get a common denominator:
\[
\frac{1}{1} + \frac{4}{c} = \frac{c + 4}{c}
\]

### Step 3: Combine the fraction
Now we can write the entire expression as:
\[
\frac{\frac{c^3 + 64}{c^2}}{\frac{c + 4}{c}}
\]

### Step 4: Simplify the complex fraction
When dividing by a fraction, multiply by its reciprocal:
\[
\frac{c^3 + 64}{c^2} \times \frac{c}{c + 4} = \frac{(c^3 + 64) \cdot c}{c^2 \cdot (c + 4)}
\]

Cancel out one \( c \) from the numerator and denominator:
\[
\frac{(c^3 + 64)}{c(c + 4)}
\]

### Step 5: Factor the numerator
The numerator, \( c^3 + 64 \), is a sum of cubes:
\[
c^3 + 64 = (c + 4)(c^2 - 4c + 16)
\]

So the expression becomes:
\[
\frac{(c + 4)(c^2 - 4c + 16)}{c(c + 4)}
\]

Cancel out the \( c + 4 \) terms:
\[
\frac{c^2 - 4c + 16}{c}
\]

### Final Answer:
The simplified expression is:
\[
\frac{c^2 - 4c + 16}{c}
\]

Would you like any more details on this solution, or have any questions?

### Here are 5 related questions to deepen your understanding:
1. How do you simplify expressions involving complex fractions?
2. What is the formula for factoring a sum of cubes?
3. How can common denominators be used when simplifying algebraic fractions?
4. Why is it important to factor terms before canceling in algebraic expressions?
5. How does multiplying by the reciprocal help in simplifying division of fractions?

### Tip:
Always check for factorizable patterns like difference of squares, sum of cubes, or other common identities when simplifying algebraic expressions.

Simplify the expression: (c + 64/c^2) / (1 + 4/c)

Learn how to simplify the complex fraction (c + 64/c^2) / (1 + 4/c) step by step. This problem involves algebraic concepts like factoring, simplifying complex fractions, and applying the sum of cubes formula. Suitable for students in Grades 9-12.

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