Problem 30 - Intermediate Algebra

Question

Simplify each of the numerical expressions. $$ -7^{2}+5^{2} $$

Step-by-Step Solution

Answer

The simplified expression is 74.

Step 1: Understanding the Expression

First, identify the expression given, which is \[ -7^{2} + 5^{2} \].This involves two components: 1. The square of -7, 2. The square of 5, and 3. Their sum.

Step 2: Calculate the Square of Negative Seven

The next step is to square the number -7. When you square a negative number, the result is positive:\[ (-7)^{2} = 49 \].

Step 3: Calculate the Square of Five

Now, calculate the square of 5:\[ 5^{2} = 25 \].

Step 4: Substitute the Squared Values

Substitute the squared values back into the original expression:\[ 49 + 25 \].

Step 5: Simplify the Expression

Finally, add the two squares together:\[ 49 + 25 = 74 \].Therefore, the simplified expression is 74.

Key Concepts

Squaring Numbers

Squaring a number simply means multiplying the number by itself. This is a mathematical operation frequently needed to simplify expressions. Whether the number is positive or negative, squaring it will yield a positive result.
For example:

• When squaring a positive number like 5, you calculate: \( 5 \times 5 = 25 \).
• When squaring a negative number such as -7, you still calculate: \( (-7) \times (-7) = 49 \).

The reason it's positive is due to the fact that a negative multiplied by a negative is positive. Remember, no matter the sign, squaring results in a positive value. Understanding this will make simplifying expressions more manageable.

Order of Operations

Correctly applying the order of operations is crucial when simplifying mathematical expressions. Known by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), this rule dictates the sequence in which operations are to be performed to ensure accuracy.
For example, in the expression \[ -7^{2} + 5^{2} \], follow these steps:

• First, handle any exponents (in this case, the squares).
• Then proceed with addition or subtraction if it's a part of the expression.

By following these steps, you ensure that each component of the expression is simplified correctly in a step-wise manner. Not applying these properly can easily lead to miscalculations, so it's important to practice this order until it becomes second nature.

Mathematical Expressions

Mathematical expressions combine numbers and operations in a way that represents a problem to be solved. Understanding the components of an expression is critical for simplification. Breaking down the expression into manageable parts is key.
Consider \[ -7^{2} + 5^{2} \]:

• Recognize each part: \( -7^{2} \) is one component, and \( 5^{2} \) is another.
• Simplify each term individually to see the sum that results.

Acting methodically ensures a clear path from complex expressions to simple numerical solutions. This clarity helps in avoiding mistakes and achieving accurate results. Mathematical expressions can often be intimidating, but with practice, they become easier to interpret and manage.