Problem 55 - Elementary Algebra

Question

For the following problems, evaluate each expression. If the expression does not represent a real number, write "not a real number." $$ -(-\sqrt{9}) $$

Step-by-Step Solution

Answer

Answer: The value of the expression \(-(-\sqrt{9})\) is 3.

Step 1: Identify the double negative

The expression is given by \(-(-\sqrt{9})\). Note that there are two negative signs applied on the square root of 9. Two negative signs will result in a positive sign.

Step 2: Compute the square root

Let's find the square root of 9 first: \(\sqrt{9} = 3\), since 3 squared is equal to 9.

Step 3: Apply the double negative

Now that we know \(\sqrt{9} = 3\), we need to apply the double negative to this value: \(-(-3)\). Two negatives make a positive, so the result is \(-(-3) = 3\).

Step 4: Write the final answer

The expression is equal to 3.

Key Concepts

Square Root

The square root of a number is a value that, when multiplied by itself, gives the original number. In mathematical terms, the square root of a number \( x \) is a number \( y \) such that \( y^2 = x \). For instance, when we evaluate the square root of 9, we are looking for a number that, when squared, equals 9. The number 3 satisfies this condition because \( 3^2 = 9 \). Hence, \( \sqrt{9} = 3 \).

• Symbol: The square root symbol is \( \sqrt{\cdot} \).
• Positive value: We typically use the non-negative square root when talking about square roots.
• Examples: \( \sqrt{16} = 4 \), \( \sqrt{25} = 5 \).

The concept of the square root is essential in algebra and higher math, making it possible to reverse the process of squaring a number.

Double Negative

A double negative occurs when two negative signs are combined in a mathematical expression. Two negative signs can cancel each other out, leading to a positive result. For example, in the expression \( -(-\sqrt{9}) \), the double negative results from the negative sign outside the parenthesis and the negative sign before the square root.

• Cancellation: Two negatives result in a positive.
• Example: \( -(-3) = 3 \).
• Application: Useful in simplifying expressions involving multiple operations.

Understanding how double negatives work is crucial for correctly interpreting and evaluating mathematical expressions. It helps in transforming expressions to their simplest form.

Real Numbers

Real numbers are the vast set of numbers that include all rational and irrational numbers. They can be found on the number line and encompass a range of numbers like whole numbers, fractions, and decimals. Real numbers include:

• Natural numbers: 1, 2, 3, …
• Integers: … -3, -2, -1, 0, 1, 2, 3, …
• Rational numbers: numbers that can be expressed as a fraction (e.g., 1/2, -4/5)
• Irrational numbers: numbers that cannot be expressed as a simple fraction (e.g., \( \sqrt{2} \), π)

Real numbers are everywhere on the number line and can be used in all basic arithmetic operations. When asked if an expression is a real number, it means determining whether the result can exist as a point on this continuum.

Mathematical Operations

Mathematical operations are the processes we use to evaluate expressions. They include addition, subtraction, multiplication, division, and more complex operations like finding roots and powers. In the expression \( -(-\sqrt{9}) \), we are dealing with several layers of operations:

• Finding the square root \( \sqrt{9} \).
• Applying the negative sign (first operation) \( -\sqrt{9} \).
• Applying another negative sign (second operation) \( -(-\sqrt{9}) \).

By following the correct order of operations, known as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), we ensure accurate results. Mastery of these operations simplifies problem-solving in mathematics and prepares us for tackling more complex expressions.