Question

find the root. assume that all variables represent nonnegative real numbers.
(sqrt4{256x^8})
select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. (sqrt4{256x^8} = square)
b. the root is not a real number.

Explanation:

Step1: Simplify the constant term

We know that \(256 = 4^4\), so \(\sqrt[4]{256}=\sqrt[4]{4^4} = 4\).

Step2: Simplify the variable term

For the variable part \(x^8\), using the property of radicals \(\sqrt[n]{a^m}=a^{\frac{m}{n}}\), we have \(\sqrt[4]{x^8}=x^{\frac{8}{4}}=x^2\).

Step3: Combine the results

Multiply the simplified constant and variable parts together. So \(\sqrt[4]{256x^8}=\sqrt[4]{256}\times\sqrt[4]{x^8}=4\times x^2 = 4x^2\).

Answer:

\(4x^2\)