Given \( x>0 \), simplify \( \sqrt{100 x^{8}} \) completely.

To simplify \( \sqrt{100 x^{8}} \), we start by breaking it down. The square root of a product can be separated into the product of the square roots: \[ \sqrt{100 x^{8}} = \sqrt{100} \cdot \sqrt{x^{8}}. \] Now, simplifying each part, we have \( \sqrt{100} = 10 \) since \( 10 \times 10 = 100 \). For \( \sqrt{x^{8}} \), we can apply the property of exponents, which gives us \( \sqrt{x^{8}} = x^{4} \), because taking the square root of \( x^{8} \) means dividing the exponent by 2. Putting it all together, we find: \[ \sqrt{100 x^{8}} = 10 \cdot x^{4} = 10x^{4}. \] So, the fully simplified form is \( 10x^{4} \).