Question

use the product rule to multiply
\\(\sqrt5{81} \cdot \sqrt5{3}\\)
\\(\sqrt5{81} \cdot \sqrt5{3} = \square\\)
(type an exact answer, using radicals as needed. simplify your answer.)

Explanation:

Step1: Recall the product rule for radicals

The product rule for radicals states that for any real numbers \(a\) and \(b\) and positive integer \(n\), \(\sqrt[n]{a}\cdot\sqrt[n]{b}=\sqrt[n]{a\cdot b}\). So we can apply this rule to \(\sqrt[5]{81}\cdot\sqrt[5]{3}\).
\[
\sqrt[5]{81}\cdot\sqrt[5]{3}=\sqrt[5]{81\times3}
\]

Step2: Calculate the product inside the radical

Calculate \(81\times3 = 243\). Now we have \(\sqrt[5]{243}\).

Step3: Simplify the fifth - root

We know that \(243 = 3^5\) (since \(3\times3\times3\times3\times3=243\)). So \(\sqrt[5]{243}=\sqrt[5]{3^5}\). And by the property of radicals \(\sqrt[n]{x^n}=x\) when \(n\) is odd (here \(n = 5\) which is odd) and \(x = 3\), we get \(\sqrt[5]{3^5}=3\).

Answer:

\(3\)