Problem 44 - Intermediate Algebra

Question

Simplify each of the numerical expressions. $$ (4-9)^{2} $$

Step-by-Step Solution

Answer

The simplified expression is 25.

Step 1: Evaluate the Expression Inside the Parentheses

The expression inside the parentheses is \(4 - 9\). Calculate this by subtracting 9 from 4, which gives you \(-5\).

Step 2: Apply the Exponent

The expression is now \((-5)^2\). When squaring a number, you multiply it by itself. So, \((-5)^2 = (-5) \times (-5)\).

Step 3: Simplify the Result

Multiply \(-5\) by \(-5\), which equals \(25\). This is because the product of two negative numbers is a positive number.

Key Concepts

Understanding the Order of Operations

The order of operations is essential when simplifying numerical expressions. It ensures that everyone interprets expressions the same way, leading to consistent results. The standard order is summarized by the acronym PEMDAS:

• P: Parentheses first
• E: Exponents (i.e., powers and roots, etc.)
• M: Multiplication
• D: Division
• A: Addition
• S: Subtraction

Generally, operations are performed from left to right. In our example, \((4 - 9)^2\), we first worked on the parentheses because that is the first step in PEMDAS. After simplifying inside the parentheses, we moved on to the exponentiation.

Grasping Exponents

Exponents are a way of expressing repeated multiplication of the same number by itself. The number being multiplied is called the base, and the exponent denotes the number of times the base is used as a factor. For example, in \(5^3\), 5 is the base and 3 is the exponent, meaning \(5 \times 5 \times 5\).

In the exercise \((-5)^2\), the base is \(-5\) and the exponent is 2. Hence, \((-5) \times (-5)\). Understanding this allows you to correctly simplify the expression by following the exponentiation rules. Remember that an exponent applies to the entire base, including its sign, if it’s negative.

Working with Negative Numbers

Negative numbers are less than zero and represented with a minus sign in front. They can be tricky, especially when performing operations like multiplication or dealing with exponents.

When multiplying two negative numbers, the result is positive. This is why in our exercise, \((-5) \times (-5) = 25\). This rule might seem counterintuitive at first, but it will make sense with practice. Another important point to remember is that \((-x)^2\) is not the same as \(-x^2\). The former means multiplying \(-x\) with itself, while the latter only squares the x, leaving a negative sign outside. To get comfortable with negative numbers, regular practice involving different operations helps in mastering the concept.