Problem 32 - Elementary Algebra

Question

Simplify. $$ \text { (12) } 2-(-23) 2 $$

Step-by-Step Solution

Answer

70

Step 1: Identify the expression

We need to simplify the expression \((12) 2 - (-23) 2\). This involves basic arithmetic operations.

Step 2: Perform the multiplication

Multiply the numbers in the parentheses with the number outside. This gives us two separate products: \(12 \times 2 = 24\) and \(-23 \times 2 = -46\).

Step 3: Handle the subtraction of a negative

When you subtract a negative number, you are essentially adding its positive counterpart. So \(24 - (-46)\) becomes \(24 + 46\).

Step 4: Perform the addition

Add 24 and 46 to get the final simplified result. \(24 + 46 = 70\).

Key Concepts

Basic Arithmetic Operations

Understanding basic arithmetic operations is crucial when simplifying expressions. They form the foundation of mathematics and include operations such as addition, subtraction, multiplication, and division.

These operations are used to simplify numbers and expressions in a variety of ways. Let’s break them down:

• Addition helps combine numbers.
• Subtraction finds the difference between numbers.
• Multiplication involves repeated addition.
• Division splits a number into equal parts.

In this exercise, you'll mainly focus on multiplication, subtraction, and addition. Each operation plays a role in simplifying the given expression.

Multiplication

Multiplication is the arithmetic operation of scaling one number by another. In this problem, we multiply numbers inside parentheses by those outside them.

Here's how you do it:

• Multiply 12 by 2: \( 12 \times 2 = 24 \)
• Multiply -23 by 2: \( -23 \times 2 = -46 \)

This results in two separate calculations. Multiplying negative numbers by positive ones results in a negative product, which is important for the next steps.

Subtraction of Negative Numbers

Subtracting negative numbers can sometimes be tricky, but it's a lot like addition. When you come across a negative sign in front of a negative number, it becomes positive.

So, if you see

• \( 24 - (-46) \),

think of it as adding the positive of 46:

• \( 24 + 46 \).

This change is due to the fact that subtracting a negative is the same as adding its positive counterpart. Remember this rule for quick simplification.

Addition

Addition combines numbers to form a larger amount. It’s straightforward once you’ve prepared the numbers by completing any necessary operations like multiplication or handling negatives.

In this exercise, after converting the subtraction of a negative to addition, you're left with:

• \( 24 + 46 \).

Simply add the numbers together:

• \( 24 + 46 = 70 \).

This final step gives you the simplified result. Addition is not just putting numbers together, but also understanding the outcomes of prior operations to achieve the correct result.