Question

simplifying a radical expression with an even exponent
simplify.
\\(\sqrt{8y^{12}}\\)
assume that the variable \\(y\\) represents a positive real number.

Explanation:

Step1: Factor the radicand

We can factor \(8y^{12}\) as \(4\times2\times y^{12}\), where \(4\) is a perfect square and \(y^{12}\) is a perfect square (since the exponent \(12\) is even). So, \(\sqrt{8y^{12}}=\sqrt{4\times2\times y^{12}}\).

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

Applying the property, we get \(\sqrt{4\times2\times y^{12}}=\sqrt{4}\times\sqrt{2}\times\sqrt{y^{12}}\).

Step3: Simplify each square root

We know that \(\sqrt{4} = 2\) and for \(\sqrt{y^{12}}\), since \(y\) is a positive real number, \(\sqrt{y^{12}}=y^{\frac{12}{2}} = y^{6}\) (using the property \(\sqrt[n]{x^{m}}=x^{\frac{m}{n}}\) for \(x\geq0\)). So, substituting these values, we have \(2\times\sqrt{2}\times y^{6}\).

Step4: Rearrange the terms

Rearranging the terms, we get \(2y^{6}\sqrt{2}\).

Answer:

\(2y^{6}\sqrt{2}\)