Problem 109 - Intermediate Algebra

Question

Choose the correct letter for each exercise. Letters will be used more than once. No pencil is needed. Just think about the meaning of each expression. \(A=2, B=-2, C=\) not a real number $$ (-4)^{1 / 2} $$

Step-by-Step Solution

Answer

C, as the expression is not a real number.

Step 1: Identify the Expression

The expression we need to solve is \((-4)^{1/2}\). This is the same as finding the square root of \(-4\).

Step 2: Understand Square Root of a Negative Number

In real numbers, the square root of a negative number is not defined. The reason is that there is no real number which, when squared, gives a negative result. Therefore, \((-4)^{1/2}\) does not produce a real number.

Step 3: Assign the Correct Letter

According to the given options, \((A=2, B=-2, C=\text{not a real number})\), the result of \((-4)^{1/2}\) corresponds to option C. This means that the expression results in a complex number, which is not considered a real number.

Key Concepts

Square Root

The square root, often denoted as \( \sqrt{ } \), is a mathematical operation that finds a number which, when multiplied by itself, equals the original number. For example, the square root of 4 is 2 since \(2 \times 2 = 4\). This is straightforward when dealing with positive numbers or zero. However, taking the square root of a negative number brings us into the realm of complex numbers.
In real numbers, a negative number does not have a square root because there is no real number which, when squared, gives a negative outcome. This restriction is why, in the original exercise, \((-4)^{1/2}\) is labeled as "not a real number". Mathematicians have extended numbers to include imaginary numbers to handle such cases. An imaginary unit, denoted as \(i\), is defined as \(i^{2} = -1\). Therefore, the square root of \(-4\) would normall be written as \(2i\) in the complex plane.

Negative Numbers

Negative numbers are values less than zero, depicted with a minus sign (-). They represent opposites, such as debts in financial terms or below-zero temperatures in weather terms. When it comes to operations involving negative numbers, certain rules apply.
For instance, multiplying two negative numbers results in a positive number. This concept aligns with the rule \((-1) \times (-1) = 1\). However, when dealing with the square roots or powers of negative numbers, things become more complex because the square of any real number, positive or negative, results in a positive number. Hence, you cannot easily determine a real number for the square root of a negative number, as expressed in the context of the given problem. This limitation leads us to explore complex numbers.

Real Numbers

Real numbers are the set of numbers that can be found on the number line, encompassing all rational and irrational numbers. They include integers (like -3, 0, and 5), fractions, and decimals, both finite and infinite (like 22/7 or \(\pi\)).
What distinguishes real numbers is that they do not include imaginary numbers. The definition of real numbers restricts their domain to numbers that can represent measurable quantities. Any number that requires the use of the imaginary unit \(i\) goes beyond real numbers, meaning negative square roots would not fall under real numbers.
Understanding the distinction between real and complex numbers is crucial in mathematics, as they govern the solutions to polynomial equations and provide a framework within which most mathematical operations we commonly use are valid.