Question

simplify.
\sqrt{50w^{8}}
assume that the variable w represents a positive real number.

Explanation:

Step1: Factor the radicand

We can rewrite \(50w^8\) as a product of a perfect square and another number. First, factor \(50\) into \(25\times2\) and \(w^8=(w^4)^2\). So, \(\sqrt{50w^8}=\sqrt{25\times2\times(w^4)^2}\).

Step2: Use the property of square roots

The property \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) (for \(a\geq0,b\geq0\)) allows us to split the square root: \(\sqrt{25\times2\times(w^4)^2}=\sqrt{25}\times\sqrt{(w^4)^2}\times\sqrt{2}\).
We know that \(\sqrt{25} = 5\) and \(\sqrt{(w^4)^2}=w^4\) (since \(w\) is a positive real number, the square root of a square is the number itself).
So, substituting these values in, we get \(5\times w^4\times\sqrt{2}=5w^4\sqrt{2}\).

Answer:

\(5w^4\sqrt{2}\)