Simplify the expression $\frac{4^{x+1} \times 8^{2x}}{2^{x+3}}$.

1. Problem Description

Simplify the expression $\frac{4^{x+1} \times 8^{2x}}{2^{x+3}}$.

2. Solution Steps

First, rewrite the terms using the same base, which is

2. We have $4 = 2^2$ and $8 = 2^3$. Thus:

$4^{x+1} = (2^2)^{x+1} = 2^{2(x+1)} = 2^{2x+2}$

$8^{2x} = (2^3)^{2x} = 2^{6x}$

Now we can rewrite the expression as:

$\frac{2^{2x+2} \times 2^{6x}}{2^{x+3}}$

Using the property $a^m \times a^n = a^{m+n}$, we have:

$2^{2x+2} \times 2^{6x} = 2^{2x+2+6x} = 2^{8x+2}$

Thus, the expression becomes:

$\frac{2^{8x+2}}{2^{x+3}}$

Using the property $\frac{a^m}{a^n} = a^{m-n}$, we have:

$\frac{2^{8x+2}}{2^{x+3}} = 2^{(8x+2)-(x+3)} = 2^{8x+2-x-3} = 2^{7x-1}$

3. Final Answer

$2^{7x-1}$