Problem 68 - Algebra for College Students

Question

Simplify each of the numerical expressions. $$3[4(6+7)]+2[3(4-2)]$$

Step-by-Step Solution

Answer

The simplified expression is 168.

Step 1: Simplify inside the innermost parentheses

Begin by solving the expressions within the innermost parentheses. First is \(6 + 7 = 13\),and then \(4 - 2 = 2\).Substitute these results into the expression: \[3[4(13)]+2[3(2)]\]

Step 2: Handle the brackets

With the simplified parentheses within brackets, perform the multiplication inside each bracket. First, calculate inside the brackets: \(4 \times 13 = 52\) and \(3 \times 2 = 6\).Substitute these into the expression: \[3[52] + 2[6]\]

Step 3: Multiply the bracket results by the numbers outside

Multiply the results from the brackets with the numbers outside the brackets. \(3 \times 52 = 156\) and \(2 \times 6 = 12\).This gives: \[156 + 12\]

Step 4: Add the final results

Add the results from the previous step: \(156 + 12 = 168\).

Key Concepts

Simplifying Expressions

When we talk about simplifying expressions, we're referring to reducing a mathematical expression to its simplest form. This involves combining like terms and performing arithmetic operations to identify a single, concise result. In the given exercise, the expression is composed of multiple operations and numbers grouped within brackets and parentheses. To simplify:

• First, solve inside the innermost parentheses or brackets using basic arithmetic.
• Then, move outward, simplifying progressively larger groups of numbers and operations.
• Finally, combine everything into one simplified expression.

By following these steps, complex expressions can be made easier to handle, and results become clearer.

Order of Operations

The order of operations is crucial when resolving expressions with multiple arithmetic operations—each operation must be performed in the correct order to accurately solve the expression. An easy way to remember the order is by using the acronym PEMDAS:

• Parentheses first.
• Exponents (i.e., powers and square roots, etc.) come next.
• Multiplication and Division, from left to right, after that.
• Lastly, Addition and Subtraction, also from left to right.

This order ensures expressions are simplified properly. Following these rules, we calculated operations within the innermost parentheses first, and then proceeded outwards. This methodical approach is vital for arriving at the correct answer.

Algebra for Beginners

For beginners, algebra can seem intimidating, but it's a logical and step-by-step process. The key is understanding how to approach each part of an algebraic expression.

• Recognize and apply the order of operations, which guides you in performing arithmetic in the correct sequence.
• Look for like terms or operations that can be grouped together or simplified.
• Tackle expressions one step at a time, breaking them into smaller, manageable parts.

Developing a familiarity with these processes builds a strong foundation in algebra, enabling anyone to solve even more complicated problems with confidence and ease. Remember, practice makes perfect, and understanding these basics will make algebra a powerful tool in your math toolkit.