Problem 47 - Elementary Algebra

Question

For the following problems, evaluate each expression. If the expression does not represent a real number, write "not a real number." $$ -\sqrt{100} $$

Step-by-Step Solution

Answer

Question: Evaluate the expression: \( - \sqrt{100} \) Answer: The evaluated expression is -10.

Step 1: Determine the square root of 100

To find the square root of 100, we need to think of a number that multiplies by itself to produce 100. We know that \(10*10 = 100\), so the square root of 100 is 10.

Step 2: Apply the negative sign

Since the expression is given as \(- \sqrt{100}\), we must apply the negative sign to the square root of 100 we calculated in the previous step. Thus, \(-10\) is the final result of the given expression.

Step 3: Confirm if the result is a real number

The result we obtained is \(-10\), which is a real number. Therefore, there is no need to write "not a real number" for this expression. The final answer is \(-10\).

Key Concepts

Real Numbers

Real numbers are a set of numbers that include all rational and irrational numbers. When we say a number is real, it means it can be placed on the number line without any imaginary components. Here are the types of numbers included under real numbers:

• Rational numbers: These are numbers that can be expressed as fractions or quotients of integers. Examples include 1/2, 4, and -3.
• Irrational numbers: These numbers cannot be expressed as a simple fraction. Examples include \( \pi \) and \( \sqrt{2} \).

The expression \(-10\) from our exercise is a real number because it is a rational number placed directly on the number line. No imaginary parts are involved. Understanding real numbers helps us evaluate basic expressions and recognize whether our results are real or not.

Negative Numbers

Negative numbers are numbers less than zero, represented with a minus sign. They are an essential part of real numbers. In the context of this exercise, we evaluated the expression \(- \sqrt{100}\), which involves a negative sign before the square root.

• A negative number, like \(-10\), indicates a value left of zero on the number line.
• Negative numbers are often used in real-world contexts, like temperatures below freezing or debts.

When applying a negative sign to a number like \( \sqrt{100} = 10 \), it transforms it to \(-10\). This operation doesn't make the expression complex or non-real, but rather shifts the value in the opposite direction on the number line.

Evaluating Expressions

Evaluating expressions means finding the value of a mathematical phrase. In our case, we evaluated \(- \sqrt{100}\) by following straightforward steps. Let's see how: To evaluate, start with finding the square root. Here, \( \sqrt{100} \) resolves to 10 since multiplying 10 by itself gives 100.

• Calculate the square root: Identify the number whose square is the original number. For 100, this is 10.
• Apply operations: Add any additional operations given in the expression. Here, apply a negative sign to convert 10 to \(-10\).

Finally, confirm whether the resulting number, \(-10\), is a real number – it is! Evaluating expressions like this helps streamline problem-solving in algebra and beyond.