Question

question
assuming ( x ) and ( y ) are both positive, write the following expression in simplest radical form.
( sqrt{125x^4y^6} )
answer
attempt 1 out of 2
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Explanation:

Step1: Factor the radicand

We can factor \(125x^{4}y^{6}\) as \(125 = 5^3\), \(x^{4}=(x^{2})^{2}\), and \(y^{6}=(y^{3})^{2}\). So, \(\sqrt{125x^{4}y^{6}}=\sqrt{5^{3}\times x^{4}\times y^{6}}\).

Step2: Use the property of square roots \(\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}\) (\(a\geq0,b\geq0\))

We can split the square root as follows: \(\sqrt{5^{3}\times x^{4}\times y^{6}}=\sqrt{5^{2}\times5\times x^{4}\times y^{6}}\). Then, using the property \(\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}\), we get \(\sqrt{5^{2}}\times\sqrt{5}\times\sqrt{x^{4}}\times\sqrt{y^{6}}\).

Step3: Simplify each square root

We know that \(\sqrt{5^{2}} = 5\), \(\sqrt{x^{4}}=x^{2}\) (since \(x\) is positive), and \(\sqrt{y^{6}}=y^{3}\) (since \(y\) is positive). So, substituting these values in, we have \(5\times x^{2}\times y^{3}\times\sqrt{5}\).

Step4: Rearrange the terms

Rearranging the terms, we get \(5x^{2}y^{3}\sqrt{5}\).

Answer:

\(5x^{2}y^{3}\sqrt{5}\)