Quick Solution

To find the arithmetic mean of two numbers, you add the numbers together and divide the sum by 2. This problem requires applying that formula and then using the rules of exponents to simplify the result.

1. Apply the Arithmetic Mean Formula

The arithmetic mean of two numbers, 'a' and 'b', is given by the formula (a + b) / 2.

Applying this to the given numbers, $2^{10}$ and $2^{20}$:

$Mean = \frac{2^{10} + 2^{20}}{2}$

2. Simplify the Expression Using Exponent Rules

We can split the fraction and simplify each term individually. We use the exponent rule $\frac{x^a}{x^b} = x^{a-b}$, remembering that 2 is the same as $2^1$.

$Mean = \frac{2^{10}}{2} + \frac{2^{20}}{2}$

$Mean = \frac{2^{10}}{2^1} + \frac{2^{20}}{2^1}$

$Mean = 2^{(10-1)} + 2^{(20-1)}$

$Mean = 2^9 + 2^{19}$

Final Answer: The arithmetic mean of $2^{10}$ and $2^{20}$ is $2^9 + 2^{19}$.