Question

simplify the radical expression.
\sqrt3{x^{7}}

Explanation:

Step1: Rewrite the exponent

We can express the exponent \(7\) as a sum of a multiple of \(3\) and a remainder. Since \(7 = 3\times2+1\), we have \(x^{7}=x^{3\times2 + 1}\).

Step2: Use the property of exponents and radicals

Recall that \(\sqrt[n]{a^{m}}=a^{\frac{m}{n}}\) and \(a^{mn}=(a^{m})^{n}\). So, \(\sqrt[3]{x^{7}}=\sqrt[3]{x^{3\times2+1}}=\sqrt[3]{(x^{2})^{3}\times x}\).
Using the property \(\sqrt[n]{ab}=\sqrt[n]{a}\times\sqrt[n]{b}\) (for \(a,b\geq0\) when \(n\) is even, and for all real \(a,b\) when \(n\) is odd), we get \(\sqrt[3]{(x^{2})^{3}\times x}=\sqrt[3]{(x^{2})^{3}}\times\sqrt[3]{x}\).
Since \(\sqrt[3]{(x^{2})^{3}} = x^{2}\) (because \(\sqrt[n]{a^{n}}=a\) when \(n\) is odd), the expression simplifies to \(x^{2}\sqrt[3]{x}\).

Answer:

\(x^{2}\sqrt[3]{x}\)