The problem is to evaluate the expression $2^{34} - 2^{33}$.

1. Problem Description

The problem is to evaluate the expression $2^{34} - 2^{33}$.

2. Solution Steps

We can factor out $2^{33}$ from the expression.

$2^{34} - 2^{33} = 2^{33} \cdot 2^1 - 2^{33} \cdot 1$

$2^{34} - 2^{33} = 2^{33} (2 - 1)$

$2^{34} - 2^{33} = 2^{33} (1)$

$2^{34} - 2^{33} = 2^{33}$

Now we need to calculate $2^{33}$.

$2^{10} = 1024 \approx 10^3 = 1000$

$2^{33} = 2^{30} \cdot 2^3 = (2^{10})^3 \cdot 8 = (1024)^3 \cdot 8$

Since we cannot use a calculator, we can approximate $(1024)^3$ as $(1000)^3 = 10^9$. So, we get approximately $8 \cdot 10^9$.

If we need a more accurate value:

$2^{33} = 8,589,934,592$

3. Final Answer

$2^{33} = 8589934592$