We are tasked with simplifying the following expression:

$\frac{2^{2024} - 2^{2020}}{2^{2020} - 2^{2016}}$

Step 1: Factor the numerator and the denominator

Both terms in the numerator $2^{2024} - 2^{2020}$ and the denominator $2^{2020} - 2^{2016}$ can be factored by taking out the smallest power of 2 from each.

Numerator:

$2^{2024} - 2^{2020} = 2^{2020}(2^{4} - 1)$

Denominator:

$2^{2020} - 2^{2016} = 2^{2016}(2^{4} - 1)$

Step 2: Substitute factored forms into the expression

The expression becomes: $\frac{2^{2020}(2^4 - 1)}{2^{2016}(2^4 - 1)}$

Step 3: Simplify

Cancel out the common factor $(2^4 - 1)$ (since $2^4 - 1 = 16 - 1 = 15$) and simplify the powers of 2: $\frac{2^{2020}}{2^{2016}} = 2^{2020 - 2016} = 2^4$

Step 4: Compute $2^4$

$2^4 = 16$

Final Answer:

$\boxed{16}$

Let me know if you want a step-by-step breakdown of any part!

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Related Questions:

1. How do you factorize exponential expressions efficiently?
2. What are common patterns in expressions involving powers of 2?
3. Can you explain the properties of exponents used here?
4. What are practical applications of such exponential simplifications?
5. How would the solution change if additional terms were involved?

Tip:

When simplifying expressions with powers of 2, always look for common factors and apply the properties of exponents systematically.