Question

1. write the expression in simplest form. write your answer as a radical expression.

\sqrt4{567} = \square

Explanation:

Step1: Prime - factorize the number inside the radical

Prime - factorize \(567\): \(567 = 3^{4}\times7\).

Step2: Use the property of the fourth - root \(\sqrt[4]{ab}=\sqrt[4]{a}\cdot\sqrt[4]{b}\) (\(a = 3^{4}\), \(b = 7\))

We know that \(\sqrt[4]{3^{4}\times7}=\sqrt[4]{3^{4}}\cdot\sqrt[4]{7}\).
Since \(\sqrt[4]{3^{4}}=3\) (because if \(y = \sqrt[n]{x^{n}}\), when \(n\) is even and \(x\geq0\), \(y=x\)), then \(\sqrt[4]{567}=3\sqrt[4]{7}\).

Answer:

\(3\sqrt[4]{7}\)