Question

simplify.
\sqrt{18y^{8}}
assume that the variable y represents a positive real number.

Explanation:

Step1: Factor the radicand

We can factor \(18y^8\) as \(9\times2\times y^8\), where \(9\) is a perfect square and \(y^8=(y^4)^2\) is also a perfect square. So, \(\sqrt{18y^8}=\sqrt{9\times2\times y^8}\).

Step2: Use the property of square roots

Using the property \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) (where \(a\geq0\) and \(b\geq0\)), we can split the square root: \(\sqrt{9\times2\times y^8}=\sqrt{9}\times\sqrt{2}\times\sqrt{y^8}\).

Step3: Simplify each square root

We know that \(\sqrt{9} = 3\) and \(\sqrt{y^8}=y^4\) (since \(y\) is a positive real number, we don't need to consider the absolute value). So, substituting these values in, we get \(3\times y^4\times\sqrt{2}\).

Step4: Combine the terms

Combining the terms, we have \(3y^4\sqrt{2}\).

Answer:

\(3y^4\sqrt{2}\)