Evaluate each expression in Exercises 1–12, or indicate that the root is not a real number. −√36

Verified step by step guidance

1. Step 1: Recognize that the problem involves evaluating the square root of a number. The square root of a number is a value that, when multiplied by itself, equals the original number.
2. Step 2: Identify the number inside the square root symbol (radical). In this case, the number is 36, so we are tasked with finding √36.
3. Step 3: Determine the square root of 36. Since 6 × 6 = 36, the square root of 36 is 6. Note that the square root symbol (√) typically refers to the principal (non-negative) square root.
4. Step 4: Apply the negative sign in front of the square root. The problem specifies −√36, so we take the negative of the square root value found in Step 3.
5. Step 5: Conclude that the expression evaluates to the negative of the square root of 36. The result is a real number because the square root of 36 is defined and real.

Key Concepts

Square Roots

A square root of a number 'x' is a value 'y' such that y² = x. For example, the square root of 36 is 6, since 6² = 36. Square roots can be both positive and negative, but when referring to the principal square root, we typically consider only the non-negative value.

Negative Square Roots

When evaluating expressions involving negative square roots, such as -√x, it indicates the negative of the principal square root of 'x'. For instance, -√36 equals -6. Understanding this concept is crucial for correctly interpreting expressions that involve both square roots and negative signs.

Real and Non-Real Numbers

In algebra, real numbers include all rational and irrational numbers, while non-real numbers typically refer to complex numbers. When evaluating square roots, if the number under the root is negative, the result is not a real number. For example, √(-1) is not a real number, but rather an imaginary unit denoted as 'i'.