Question

2 find the reduced radical form of \\(\sqrt{250y^{3}z^{4}}\\)

Explanation:

Step1: Factor the radicand

First, we factor \(250y^{3}z^{4}\) into prime factors and separate the perfect - square factors.
We know that \(250 = 2\times125=2\times5^{3}\), \(y^{3}=y^{2}\times y\), and \(z^{4}=(z^{2})^{2}\).
So, \(250y^{3}z^{4}=2\times5^{3}\times y^{2}\times y\times(z^{2})^{2}\).
We can rewrite it as \(250y^{3}z^{4}=5^{2}\times5\times y^{2}\times y\times(z^{2})^{2}\).

Step2: Simplify the square root

Using the property \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) (\(a\geq0,b\geq0\)) and \(\sqrt{a^{2}} = a\) (\(a\geq0\)), we have:
\(\sqrt{250y^{3}z^{4}}=\sqrt{5^{2}\times5\times y^{2}\times y\times(z^{2})^{2}}\)
\(=\sqrt{5^{2}}\times\sqrt{y^{2}}\times\sqrt{(z^{2})^{2}}\times\sqrt{5y}\)
Since \(\sqrt{5^{2}} = 5\), \(\sqrt{y^{2}}=y\) (assuming \(y\geq0\)) and \(\sqrt{(z^{2})^{2}}=z^{2}\), we get:
\(= 5\times y\times z^{2}\times\sqrt{5y}\)
\(=5yz^{2}\sqrt{5y}\)

Answer:

\(5yz^{2}\sqrt{5y}\)