Problem 64 - Intermediate Algebra

Question

Simplify each of the numerical expressions. $$ (3 \cdot 4+2 \cdot 1)(5 \cdot 2+6 \cdot 7) $$

Step-by-Step Solution

Answer

The simplified expression is 728.

Step 1: Calculate inside the parenthesis

Start by solving each of the expressions inside the parentheses separately. For the first parenthesis:\(3 \cdot 4 + 2 \cdot 1 = 12 + 2 = 14\)For the second parenthesis:\(5 \cdot 2 + 6 \cdot 7 = 10 + 42 = 52\)

Step 2: Multiply the results

Now that you have simplified the expressions inside the parentheses, multiply the results from each parenthesis.This gives:\(14 \cdot 52\)

Step 3: Perform the multiplication

Finally, calculate the multiplication:\(14 \cdot 52 = 728\)

Key Concepts

Parentheses

In mathematical expressions, parentheses play a critical role in determining the order of operations. They are used to group parts of a numerical expression so that operations within the parentheses are performed first. This ensures clarity and avoids any ambiguity in the sequence of operations.

In the expression \((3 \cdot 4 + 2 \cdot 1)(5 \cdot 2 + 6 \cdot 7)\), each set of parentheses indicates that the calculations inside must be completed separately before any other operations, like multiplication, are performed. Here’s a simplified approach:

• Identify expressions inside each parenthesis.
• Perform operations within the parentheses:
• For example, \(3 \cdot 4 + 2 \cdot 1 = 14\)
• Also, \(5 \cdot 2 + 6 \cdot 7 = 52\)

By completing the operations within the parentheses first, we ensure the entire expression is simplified correctly and efficiently.

Multiplication

Once you have simplified the expressions inside the parentheses, multiplication becomes the next crucial step in evaluating a numerical expression. In our example, after simplifying what's inside the parentheses, you get \(14\) and \(52\).

The rule of multiplication requires you to take these two simplified results and multiply them together. Here’s how you can approach it:

• Write down the simplified numbers derived from each parenthesis and prepare to multiply them.
• Perform the multiplication: \(14 \times 52\).
• If you're unsure how to multiply such numbers directly, you can use multiplication in parts, such as distributing \(14\) over \((50 + 2)\).

This would mean calculating \(14 \times 50 + 14 \times 2\), which will also give you the final result of \(728\). This breakdown helps make sense of any complex multiplication by breaking it down into easier parts.

Numerical Expressions

Understanding and simplifying numerical expressions require a methodical approach. A numerical expression is essentially a statement involving numbers and operations, such as addition, subtraction, multiplication, or division. The main goal is to simplify these expressions to reach a single numerical outcome.

To simplify a numerical expression, you can follow these steps:

• Address operations within any existing parentheses first to understand preliminary results.
• Follow the correct order of operations, known as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
• Use basic arithmetic to further simplify, combining similar terms where necessary.

The expression \((3 \cdot 4 + 2 \cdot 1)(5 \cdot 2 + 6 \cdot 7)\) exemplifies using parentheses to direct the proper order of operations and then applying other arithmetic to simplify to a single result. Remember, accurate evaluation of numerical expressions helps in solving more complex mathematical problems efficiently.