Question

simplify the radical expression (sqrt4{1250x^2}) (the image shows the fourth - root of (1250x^{2}))

Explanation:

Step1: Factor the radicand

First, factor 1250. We know that \(1250 = 2\times625 = 2\times5^4\). So the expression becomes \(\sqrt[4]{2\times5^4\times x^2}\).

Step2: Simplify the fourth - root

Using the property of radicals \(\sqrt[n]{ab}=\sqrt[n]{a}\times\sqrt[n]{b}\) (\(a\geq0,b\geq0\)) and \(\sqrt[n]{a^n}=a\) (when \(n\) is even and \(a\geq0\)), we have:
\(\sqrt[4]{2\times5^4\times x^2}=\sqrt[4]{5^4}\times\sqrt[4]{2x^2}\)
Since \(\sqrt[4]{5^4} = 5\), the simplified form is \(5\sqrt[4]{2x^2}\). We can also rewrite \(x^2\) as \(x^{4 - 2}\), but for the fourth - root of \(x^2\), we can leave it as \(\sqrt[4]{x^2}=\sqrt{\vert x\vert}\) (because \(\sqrt[4]{x^2}=(x^2)^{\frac{1}{4}}=x^{\frac{2}{4}}=x^{\frac{1}{2}}=\sqrt{\vert x\vert}\) when considering real numbers). However, if we assume \(x\geq0\), then \(\sqrt[4]{x^2}=\sqrt{x}\). So the expression can be further written as \(5\sqrt[4]{2}\times\sqrt{\vert x\vert}\) or \(5\sqrt[4]{2x^2}\) (the simplest radical form without considering the absolute value for non - negative \(x\) is \(5\sqrt[4]{2x^2}\)).

Answer:

\(5\sqrt[4]{2x^2}\) (or \(5\sqrt{\vert x\vert}\sqrt[4]{2}\) if we separate the radicals for the variable part)