Find each root. √(-12)²

Verified step by step guidance

1. First, recognize that the expression is \( \sqrt{(-12)^2} \), which means the square root of \((-12)\) squared.
2. Recall the property that \( \sqrt{a^2} = |a| \), where \(|a|\) denotes the absolute value of \(a\).
3. Apply this property to the expression: \( \sqrt{(-12)^2} = |-12| \).
4. Calculate the absolute value of \(-12\), which is the distance from zero on the number line, so \(|-12| = 12\).
5. Therefore, the root of the expression \( \sqrt{(-12)^2} \) is \(12\).

Key Concepts

Order of Operations

The order of operations dictates the sequence in which mathematical operations are performed. Parentheses and exponents are evaluated before multiplication or square roots. In the expression √(-12)², the exponent applies first to -12, then the square root is taken.

Squaring Negative Numbers

Squaring a negative number results in a positive number because multiplying two negative factors yields a positive product. For example, (-12)² equals 144, which is positive, regardless of the negative sign inside the parentheses.

Square Root Function

The square root of a number is a value that, when squared, gives the original number. The principal square root is always non-negative. Thus, √144 equals 12, not -12, since the square root function returns the positive root by convention.