In Exercises 77–90, simplify each expression. Include absolute value bars where necessary.__⁴√y⁴

Verified step by step guidance

1. Recognize that the expression involves a fourth root: \( \sqrt[4]{y^4} \).
2. Recall the property of radicals: \( \sqrt[n]{a^n} = |a| \) when \( n \) is even.
3. Apply this property to the expression: \( \sqrt[4]{y^4} = |y| \).
4. Understand that the absolute value is necessary because the fourth root of a number should be non-negative.
5. Conclude that the simplified form of the expression is \( |y| \).

Key Concepts

Radicals

Radicals are expressions that involve roots, such as square roots or fourth roots. The notation √ or n√ indicates the n-th root of a number. Understanding how to simplify radical expressions is crucial, as it involves recognizing the relationship between exponents and roots, particularly when dealing with variables.

Exponents

Exponents represent repeated multiplication of a number by itself. For example, y⁴ means y multiplied by itself four times. When simplifying expressions with exponents, it's important to apply the laws of exponents, such as the power of a power rule, which states that (a^m)^n = a^(m*n).

Absolute Value

Absolute value measures the distance of a number from zero on the number line, regardless of direction. It is denoted by vertical bars, such as |x|. When simplifying expressions involving even roots, like the fourth root, the result must be expressed as a non-negative value, which is where absolute value becomes necessary.