Evaluate each expression in Exercises 1–12, or indicate that the root is not a real number. $\sqrt{(-17{)}^2}$

Verified step by step guidance

1. Identify the expression inside the square root: \(\sqrt{(-17)^2}\).
2. Recall that squaring a number means multiplying it by itself, so \((-17)^2 = (-17) \times (-17)\).
3. Calculate \((-17) \times (-17)\), which results in a positive number because a negative times a negative is positive.
4. Rewrite the expression as \(\sqrt{\text{positive number}}\).
5. Since the square root of a positive number is a real number, simplify \(\sqrt{(-17)^2}\) to the absolute value of \(-17\), which is \(17\).

Key Concepts

Square Root and Its Properties

The square root of a number is a value that, when multiplied by itself, gives the original number. For non-negative numbers, the square root is real and non-negative. Understanding how to simplify square roots, especially with squared terms, is essential for evaluating expressions like √(−17)^2.

Order of Operations

The order of operations dictates the sequence in which parts of a mathematical expression are evaluated. Parentheses and exponents are handled before roots. In the expression √(−17)^2, the exponent applies to −17 first, then the square root is taken, which affects the final result.

Real Numbers and Imaginary Numbers

Real numbers include all rational and irrational numbers, while imaginary numbers involve the square root of negative values. Recognizing when an expression results in a real number or when it involves imaginary numbers helps determine if the root is real or not, as asked in the question.