Question

question
simplify \\(\sqrt4{16x^{16}}\\) completely given \\(x > 0\\).

answer attempt 1 out of 4

Explanation:

Step1: Recall the nth root property

For a non - negative real number \(a\) and positive integer \(n\), \(\sqrt[n]{ab}=\sqrt[n]{a}\cdot\sqrt[n]{b}\) (when \(a\geq0,b\geq0\)) and \(\sqrt[n]{a^{m}} = a^{\frac{m}{n}}\) (when \(a\geq0\)). We can split the fourth - root of the product \(16x^{16}\) into the product of the fourth - root of \(16\) and the fourth - root of \(x^{16}\), i.e., \(\sqrt[4]{16x^{16}}=\sqrt[4]{16}\cdot\sqrt[4]{x^{16}}\).

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

We know that \(16 = 2^{4}\). Using the property \(\sqrt[n]{a^{n}}=a\) (for \(a\geq0\) and \(n\) is a positive integer), when \(n = 4\) and \(a = 2\), we have \(\sqrt[4]{16}=\sqrt[4]{2^{4}}=2\).

Step3: Simplify \(\sqrt[4]{x^{16}}\)

Using the property \(\sqrt[n]{a^{m}}=a^{\frac{m}{n}}\), here \(n = 4\) and \(m = 16\). So \(\sqrt[4]{x^{16}}=x^{\frac{16}{4}}\). Since \(\frac{16}{4}=4\), we get \(\sqrt[4]{x^{16}}=x^{4}\) (because \(x>0\), we don't have to consider the negative root).

Step4: Multiply the two simplified terms

Multiply the results from Step 2 and Step 3: \(\sqrt[4]{16}\cdot\sqrt[4]{x^{16}}=2\cdot x^{4}=2x^{4}\).

Answer:

\(2x^{4}\)