Multiple Choice
Simplify the following.
9. Roots, Radicals, and Complex Numbers
Introduction to Radical Expressions
Multiple Choice
Simplify the following.
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Verified step by step guidance1
Recognize that the expression involves the 8th root of \( (x+1)^8 \), written as \( \sqrt[8]{(x+1)^8} \). The 8th root and the 8th power are inverse operations.
Use the property of radicals and exponents: \( \sqrt[n]{a^n} = |a| \) when \(n\) is even, to simplify \( \sqrt[8]{(x+1)^8} \) to \( |x+1| \). This is because the even root of an even power results in the absolute value of the base.
Note the negative sign outside the radical, so the entire expression becomes \( -|x+1| \).
Understand that the absolute value \( |x+1| \) ensures the expression inside is always non-negative, regardless of the value of \( x \).
Therefore, the simplified form of the original expression is \( -|x+1| \).
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