Simplify the following.
${}^4\sqrt{x^4{y}^8}$

• A. $x{y}^4$
• B. $\mid y^2\mid$
• C. $\(\left\)|xy^2\(\right\)|$
• D. $\(\left\)|y^4\(\right\)|$

Answer: (C)

Verified step by step guidance

1. Identify the expression to simplify: the fourth root of the product \(x^4 y^8\), which can be written as \(\sqrt[4]{x^4 y^8}\).
2. Recall that the fourth root of a product is the product of the fourth roots: \(\sqrt[4]{x^4 y^8} = \sqrt[4]{x^4} \times \sqrt[4]{y^8}\).
3. Apply the property of radicals and exponents: \(\sqrt[4]{x^4} = x^{4/4} = x^1 = x\) and \(\sqrt[4]{y^8} = y^{8/4} = y^2\).
4. Consider the domain of the variables: since we are taking an even root (fourth root), the result must be nonnegative. Therefore, use absolute value to ensure the expression is valid for all real numbers, especially for \(y^2\) which is always nonnegative but \(x\) might not be.
5. Combine the results with absolute value: the simplified expression is \(\left| x y^2 \right|\).