Question

question
simplify (sqrt4{16x^{32}}) completely given (x > 0).

Explanation:

Step1: Apply the nth root property

The nth root of a product is the product of the nth roots, so \(\sqrt[4]{16x^{32}}=\sqrt[4]{16}\cdot\sqrt[4]{x^{32}}\).

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

We know that \(16 = 2^4\), so \(\sqrt[4]{16}=\sqrt[4]{2^4}=2\) (since \(x>0\), we don't have to consider negative roots here for the real - valued simplification).

Step3: Simplify \(\sqrt[4]{x^{32}}\)

Using the property \(\sqrt[n]{a^m}=a^{\frac{m}{n}}\), for \(n = 4\) and \(m = 32\), we have \(\sqrt[4]{x^{32}}=x^{\frac{32}{4}}=x^{8}\).

Step4: Multiply the results

Multiply the results from Step 2 and Step 3: \(2\cdot x^{8}=2x^{8}\).

Answer:

\(2x^{8}\)