Given that $4(5^{x+3}) = 20^{3-x}$, evaluate $10^x$ without using a calculator.

1. Expand the terms using laws of indices to separate the constants from the terms with $x$.

$4(5^x \cdot 5^3) = \frac{20^3}{20^x}$

2. Group the terms with $x$ on one side and constants on the other.

$5^x \cdot 20^x = \frac{20^3}{4 \cdot 5^3}$

3. Simplify both sides using index laws and arithmetic.

$(5 \cdot 20)^x = \frac{8000}{4 \cdot 125} 100^x = \frac{8000}{500} 100^x = 16$

4. Express $100^x$ in terms of $10^x$ to find the final value.

$(10^2)^x = 16 (10^x)^2 = 16 10^x = \sqrt{16} = 4 \quad \text{(since } 10^x > 0 \text{)}$