Evaluate each expression in Exercises 1–12, or indicate that the root is not a real number. √(−17)^2

Verified step by step guidance

1. <insert step 1> Start by understanding the expression \( \sqrt{(-17)^2} \). This involves taking the square root of the square of -17.>
2. <insert step 2> Calculate \((-17)^2\). When you square a negative number, the result is positive because multiplying two negative numbers gives a positive result.>
3. <insert step 3> So, \((-17)^2 = 289\).>
4. <insert step 4> Now, take the square root of 289. The square root function \(\sqrt{x}\) asks what number multiplied by itself gives \(x\).>
5. <insert step 5> Determine the positive number that, when squared, equals 289.>

Key Concepts

Square Root Function

The square root function, denoted as √x, is defined as the value that, when multiplied by itself, gives x. For non-negative numbers, the square root is a real number. However, when dealing with negative numbers, the square root is not defined in the set of real numbers, leading to complex numbers instead.

Exponentiation

Exponentiation is a mathematical operation involving two numbers, the base and the exponent. In the expression (−17)^2, the base is −17 and the exponent is 2, meaning that −17 is multiplied by itself. This results in a positive value, as squaring any real number, whether positive or negative, yields a non-negative result.

Complex Numbers

Complex numbers are numbers that have a real part and an imaginary part, typically expressed in the form a + bi, where 'a' is the real part and 'bi' is the imaginary part. The square root of a negative number, such as √(−x), results in an imaginary number, indicating that the solution is not a real number but rather exists in the complex number system.