Question

use the product rule to multiply.
\\(\sqrt5{16} \cdot \sqrt5{2}\\)

Explanation:

Step1: Recall the product rule for radicals

The product rule for radicals states that \(\sqrt[n]{a} \cdot \sqrt[n]{b}=\sqrt[n]{a\cdot b}\) when \(n\) is a positive integer and \(a,b\) are real numbers for which the radicals are defined. Here, \(n = 5\), \(a=16\) and \(b = 2\).
So, \(\sqrt[5]{16}\cdot\sqrt[5]{2}=\sqrt[5]{16\times2}\)

Step2: Simplify the radicand

Calculate \(16\times2=32\). So now we have \(\sqrt[5]{32}\)

Step3: Evaluate the fifth - root

We know that \(2^5=32\), so \(\sqrt[5]{32}=\sqrt[5]{2^5}\)
By the property \(\sqrt[n]{x^n}=x\) (for real numbers \(x\) and positive integer \(n\)), when \(n = 5\) and \(x = 2\), we get \(\sqrt[5]{2^5}=2\)

Answer:

\(2\)