Problem 45 - Basic College Mathematics

Question

Simplify. $$ 8 \cdot 12-(63 \div 9+13 \cdot 3) $$

Step-by-Step Solution

Answer

50

Step 1: Simplify the Division Inside the Parentheses

First, evaluate the expression inside the parentheses by simplifying the division. Calculate \( 63 \div 9 \): \[ 63 \div 9 = 7 \]

Step 2: Simplify the Multiplication Inside the Parentheses

Next, simplify the multiplication inside the parentheses: \[ 13 \cdot 3 = 39 \]

Step 3: Add the Results Inside the Parentheses

Now, add the results from the division and multiplication: \[ 7 + 39 = 46 \]

Step 4: Simplify the Multiplication Outside the Parentheses

Calculate the multiplication outside the parentheses: \[ 8 \cdot 12 = 96 \]

Step 5: Subtract the Results From Each Other

Finally, subtract the sum inside the parentheses from the multiplication outside the parentheses: \[ 96 - 46 = 50 \]

Key Concepts

Order of Operations

The order of operations is a fundamental concept in mathematics. It helps us determine the sequence in which we solve parts of an expression. The standard order is given by the acronym PEMDAS, which stands for:

• Parentheses
• Exponents
• Multiplication and Division
• Addition and Subtraction

This means you first solve expressions in parentheses, then evaluate any exponents. Next, perform any multiplication or division from left to right, and finally, do any addition or subtraction from left to right. In the example problem, we use the order of operations to simplify the expression step by step.

Basic Arithmetic

Basic arithmetic involves the fundamental operations like addition, subtraction, multiplication, and division. Mastering these operations is crucial for solving complex mathematical problems.
In our example:

• We first use division: \( 63 \div 9 \)
• Then multiplication: \( 13 \cdot 3 \)
• Next addition: \( 7 + 39 \)
• More multiplication: \( 8 \cdot 12 \)
• Finally, subtraction: \( 96 - 46 \)

Each step builds on the last to simplify the expression fully. Practicing these operations helps in achieving accuracy and speed.

Parentheses in Math

Parentheses are important because they indicate which operations should be performed first. When simplifying mathematical expressions, always start with the operations inside parentheses.
In the given problem:

• We start by simplifying the division inside the parentheses: \( 63 \div 9 = 7 \)
• Next, we handle the multiplication inside the parentheses: \( 13 \cdot 3 = 39 \)
• Then add the results: \( 7 + 39 = 46 \)

By doing this, we ensure we follow the correct order of operations, arriving at the accurate result.

Multiplication and Division

Multiplication and division are essential parts of arithmetic. According to the order of operations, they should be performed from left to right, after parentheses and exponents.
In our problem:

• We first do the division: \( 63 \div 9 = 7 \)
• Then the multiplication inside the parentheses: \( 13 \cdot 3 = 39 \)
• Outside the parentheses, we have further multiplication: \( 8 \cdot 12 = 96 \)

These operations simplify parts of the expression step by step. Understanding and practicing multiplication and division help in handling a wide range of mathematical problems efficiently.