Simplify the Exponent Equation: (2^-4 * (1/2)^8 * 2^10) / 2^3

Solve the following problem:

$\frac{2^{-4}\cdot(\frac{1}{2})^8\cdot2^{10}}{2^3}=\text{?}$

In order to solve this problem, we'll follow these steps:

• Step 1: Simplify each component using exponent rules

• Step 2: Apply multiplication and division of powers

• Step 3: Simplify the combined expression

Now, let's work through each step:

Step 1: Simplify $(\frac{1}{2})^8$. Using the power of a fraction rule, we have:

$\left(\frac{1}{2}\right)^8 = \frac{1^8}{2^8} = \frac{1}{2^8} = 2^{-8}$

Step 2: Substitute back into the original expression:

$\frac{2^{-4} \cdot 2^{-8} \cdot 2^{10}}{2^3}$

Combine the terms in the numerator using the product of powers rule:

$2^{-4} \cdot 2^{-8} \cdot 2^{10} = 2^{-4 + (-8) + 10} = 2^{-2}$

Now the expression becomes:

$\frac{2^{-2}}{2^3}$

Apply the division of powers rule:

$\frac{2^{-2}}{2^3} = 2^{-2 - 3} = 2^{-5}$

Thus, the solution to the problem is $2^{-5}$.