To simplify $2^{10} \times 2^2$, use the rule of exponents: $a^m \times a^n = a^{m+n}$.

$2^{10} \times 2^2 = 2^{10 + 2} = 2^{12}$

Now, calculate $2^{12}$:

$2^{12} = 4096$

Thus, $2^{10} \times 2^2 = 4096$.

Would you like a more detailed explanation, or do you have any further questions?

Here are 5 related questions for further exploration:

1. What is $2^5 \times 2^3$?
2. How do you simplify $3^4 \times 3^2$?
3. What is the value of $5^6 \times 5^2$?
4. How do you simplify $(2^3)^4$?
5. What is the value of $2^7 \times 2^5$?

Tip: When multiplying powers with the same base, simply add the exponents!