Simplify the following.
${-}^8\sqrt{{(x+1)}^8}$

• A. $x-1$
• B. $-8\mid x+1\mid$
• C. $-\(\left\)|x+1\(\right\)|$
• D. $\(\left\)|x+1\(\right\)|$

Answer: (C)

Verified step by step guidance

1. Recognize that the expression involves the eighth root of \((x+1)^8\), written as \(-^8\sqrt{(x+1)^8}\).
2. Recall the property of radicals and exponents: the nth root of a number raised to the nth power simplifies to the absolute value of that number, i.e., \(\sqrt[n]{a^n} = |a|\).
3. Apply this property to simplify \(^8\sqrt{(x+1)^8}\) to \(|x+1|\).
4. Include the negative sign outside the radical, so the expression becomes \(- |x+1|\).
5. Understand that the absolute value ensures the result is non-negative inside, and the negative sign outside makes the entire expression non-positive.