Problem 46 - Elementary Algebra

Question

For the following problems, evaluate each expression. If the expression does not represent a real number, write "not a real number." $$ -\sqrt{36} $$

Step-by-Step Solution

Answer

Question: Evaluate the expression \(-\sqrt{36}\). Answer: -6

Step 1: Identify the number inside the square root

In this case, the number inside the square root is 36.

Step 2: Evaluate the square root of the number

Calculate the square root of 36: \(\sqrt{36} = 6\)

Step 3: Apply the negative sign

Now, apply the negative sign to the result: \(-\sqrt{36} = -6\)

Step 4: Check if the result is a real number

The result, -6, is a real number.

Step 5: Write the final answer

The evaluated expression, \(-\sqrt{36}\), is equal to -6.

Key Concepts

Real Numbers

Real numbers are incredibly important in mathematics as they encompass a vast array of numbers we use in everyday life. These include all rational numbers, like fractions and whole numbers, as well as irrational numbers, which cannot be expressed as simple fractions.
Real numbers are essentially any number that you can think of that isn't imaginary. They can be either positive, negative, or zero, and they include:

• Perfect squares, like 36
• Decimals
• Whole numbers
• Fractions

Additionally, real numbers can be represented on a number line, which helps in visualizing their magnitude relative to each other. Understanding real numbers is crucial because they form the foundation for more complex mathematical concepts.
In the evaluated expression \(-\sqrt{36}\), the result is \(-6\), which is a real number. This conforms to the concept that real numbers can be negative, unlike imaginary numbers which are separate from the real number system.

Negative Numbers

Negative numbers extend the number line in the opposite direction of the positive numbers and zero. They are less common in everyday counting but essential in mathematics for representing values less than zero. Negative numbers are written with a minus sign before the number, indicating value below zero.
In mathematics, negative numbers often show loss, debt, or decrease. For example:

• Debt of \(-10\)
• Temperature drop of \(-5\) degrees
• Elevation below sea level

In the given problem, after calculating \(\sqrt{36}\), we apply a negative sign, giving \(-6\). This negative sign is crucial as it determines the expression’s final result. Understanding how to work with negative numbers ensures accurate calculations in expressions and equations.

Expression Evaluation

Expression evaluation is a fundamental concept where an expression, like \( -\sqrt{36} \), is simplified to find its value. This involves a series of steps that systematically reduce complex expressions to simpler or single-number forms.
To evaluate an expression accurately:

• First, solve any operations within parentheses or radicals, such as finding a square root.
• Next, handle any exponentiation or roots that need evaluating, like \( \sqrt{36} = 6 \).
• Finally, apply any additional operations, such as multiplication, division, or sign changes. In this case, applying the negative sign gives \(-6\).

By following this order, you ensure that you are simplifying the expression accurately and adhering to the proper mathematical principles.
Understanding the concept of expression evaluation helps in resolving mathematical problems efficiently and error-free.