Question

simplify the radical expression.
\sqrt4{x^{16}y^{4}}
write your answer in the form a, \sqrt4{b} , or
expressions in x and y. use at most one radical in
value symbol in your expression for a.

Explanation:

Step1: Apply the fourth - root property

The fourth - root of a product \(\sqrt[4]{ab}=\sqrt[4]{a}\cdot\sqrt[4]{b}\) (where \(a = x^{16}\) and \(b = y^{4}\)), so \(\sqrt[4]{x^{16}y^{4}}=\sqrt[4]{x^{16}}\cdot\sqrt[4]{y^{4}}\).

Step2: Simplify \(\sqrt[4]{x^{16}}\)

Recall the property of radicals \(\sqrt[n]{a^{m}}=a^{\frac{m}{n}}\). For \(\sqrt[4]{x^{16}}\), \(n = 4\) and \(m=16\), so \(\sqrt[4]{x^{16}}=x^{\frac{16}{4}}=x^{4}\).

Step3: Simplify \(\sqrt[4]{y^{4}}\)

Using the same property \(\sqrt[n]{a^{m}}=a^{\frac{m}{n}}\) with \(n = 4\) and \(m = 4\), we get \(\sqrt[4]{y^{4}}=y^{\frac{4}{4}}=y\).

Step4: Multiply the results

Multiply the simplified forms of \(\sqrt[4]{x^{16}}\) and \(\sqrt[4]{y^{4}}\): \(x^{4}\cdot y=x^{4}y\).

Answer:

\(x^{4}y\)