Problem 62 - Intermediate Algebra

Question

Simplify each of the numerical expressions. $$ 48-(14-11)(10-6) $$

Step-by-Step Solution

Answer

The expression simplifies to 36.

Step 1: Simplify Inside the Parentheses

Start by simplifying the expression inside the parentheses: \( 14 - 11 \). This gives us \( 3 \). Now, rewrite the expression as \( 48 - 3(10-6) \).

Step 2: Continue Simplifying Inside the Parentheses

Now, simplify the remaining expression within the second parentheses: \( 10 - 6 \), which equals \( 4 \). This changes the expression to \( 48 - 3 imes 4 \).

Step 3: Perform the Multiplication

Next, multiply \( 3 \) by \( 4 \), which results in \( 12 \). The expression now becomes \( 48 - 12 \).

Step 4: Final Subtraction

Finally, subtract \( 12 \) from \( 48 \), which results in \( 36 \). This is the simplified value of the original expression.

Key Concepts

Order of Operations

When dealing with expressions in mathematics, it's crucial to follow the Order of Operations. This is a set of rules that indicate which operation to perform first in order to accurately solve the expression. Typically, you'll remember this with the acronym PEMDAS:

• Parentheses
• Exponents
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

Within each level of operations, you must work from left to right. This ensures that you simplify expressions in a standard, understandable manner. For example, in the expression \(48 - (14 - 11) \times (10 - 6)\), you begin by simplifying the operations inside each set of parentheses. Then, move on to multiplication before tackling any addition or subtraction. Following this order maintains accuracy in calculations and simplifies complex algebraic work.

Arithmetic Operations

Arithmetic operations are the fundamental building blocks of mathematics. They include addition, subtraction, multiplication, and division. Each of these operations has its role and placement when simplifying expressions.

1. **Addition and Subtraction**: These operations are usually performed towards the end of simplifying an expression. They help adjust values and can change the sign of the numbers involved.
2. **Multiplication and Division**: Typically calculated before addition and subtraction when following the Order of Operations. They serve to scale numbers up or down efficiently.

By understanding and accurately applying these basic arithmetic operations, students can tackle more complicated mathematical problems and expressions with confidence. In the example provided, the multiplication step transforms the expression \(48 - 3 \times 4\) into \(48 - 12\), demonstrating the impact of this operation.

Algebraic Expressions

Algebraic expressions may include numbers, variables, and arithmetic operations. Simplifying them involves combining like terms and following mathematical operations to reduce them to the simplest form.

- **Components of Expressions**: Algebraic expressions consist of terms, which can be constants, variables, or a combination of both. For the provided exercise, though variables are absent, understanding how to manipulate constants alone is vital.
- **Like Terms**: In the algebraic realm, "like" terms can be combined. However, in our numerical expression \(48 - (14 - 11)(10 - 6)\), we apply simplification only to numbers.

Understanding algebraic expression structure allows for simplifying numerical expressions more intuitively. Simplifying such expressions is an essential skill for progressing into more complex mathematics, making algebra less daunting. The process used in the original exercise shows the importance of simplifying and performing operations step-by-step.