Problem 75 - Elementary Algebra

Question

For exercises 15-100, evaluate. $$ (4-9)^{2}-3(-1) $$

Step-by-Step Solution

Answer

28

Step 1: Simplify Inside the Parentheses

First, solve the expression inside the parentheses: (4-9)^{2}-3(-1). Calculate the value of (4-9). 4-9=-5.

Step 2: Square the Result

Next, square the result obtained from the previous step. (-5)^{2}. Calculate (-5)^{2}. (-5)^{2}=25.

Step 3: Simplify the Multiplication

Simplify the multiplication in the expression: -3(-1). Calculate -3(-1). -3(-1) = 3.

Step 4: Combine the Results

Finally, add the results from the squared term and the multiplication: 25 + 3. Calculate 25 + 3. 25 + 3 = 28.

Key Concepts

simplifying expressions

In algebra, simplifying expressions means making them easier to understand or solve. This often involves combining like terms, which are terms that have the same variables raised to the same powers. In our exercise, the first step is simplifying inside the parentheses. We have \((4-9)\). Subtract 9 from 4, and we get \(-5\). Once simplified, the parentheses disappear in this case, leaving us with the number \(-5\). This makes the rest of the expression easier to handle.

Always simplify inside parentheses first to make your calculations straightforward.

order of operations

The order of operations is crucial for solving mathematical expressions accurately. This set of rules tells us the sequence in which operations should be performed. The common acronym to remember this is PEMDAS:

• P: Parentheses first
• E: Exponents (like squaring numbers)
• M and D: Multiplication and Division (left to right)
• A and S: Addition and Subtraction (left to right)

In our exercise, we start with the parentheses \((4-9)\), then move to the exponent by squaring \(-5\). After dealing with these, we do the multiplication \(-3(-1)\) before finally combining all the results.

squaring numbers

Squaring a number means multiplying the number by itself. This is indicated by the exponent 2. For instance, \(a^{2}\) means \(a\times a\). In our example, once we have \(-5\) from \((4-9)\), we need to square it.

To square \(-5\):
Calculate \(-5\times -5\). Remember that the negative signs cancel each other out, which makes the result positive. So, \(-5^{2}=25\). Squaring numbers is a fundamental concept and is often used in evaluating expressions and solving equations.

multiplication rules

Understanding the rules of multiplication is essential in algebra. Here are some key points:

• Positive times Positive = Positive (e.g., 3 × 2 = 6)
• Negative times Positive = Negative (e.g., -3 × 2 = -6)
• Negative times Negative = Positive (e.g., -3 × -2 = 6)

In our example, we need to multiply \(-3\) and \(-1\) after we have squared \(-5\).

Calculate \(-3 \times -1\) to get 3. This follows the rule that multiplying two negative numbers results in a positive number. Once you multiply them out, combine it with earlier results to find the final answer.