Problem 30 - Algebra 1

Question

Evaluate the expression. $$7[(18-6)-6]$$

Step-by-Step Solution

Answer

The evaluated expression is 42.

Step 1: Simplify inside of Parentheses

First, solve the operation in the parentheses. Thus, \(18 - 6 = 12\). The expression simplifies to \(7[(12) - 6]\).

Step 2: Continue to Simplify Inside of Brackets

The next step is to further simplify the expression inside the brackets. Compute \(12 - 6 = 6\). Now the expression simplifies to \(7[6]\).

Step 3: Perform Multiplication

Finally, perform the multiplication operation. Calculate \(7 \times 6 = 42\).

Key Concepts

Expressions

Expressions are combinations of numbers, operations, and sometimes variables, that represent a particular value. In this exercise, the expression given is written as \(7[(18-6)-6]\).
An expression can be quite complex, incorporating multiple operations like addition, subtraction, multiplication, and division. To solve or simplify an expression, you follow a standard set of rules known as the order of operations.
These rules help ensure that everyone evaluates the expression consistently. The acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)) is commonly used to remember the sequence.
When working with expressions, always start simplifying from the innermost operation and work your way out. This will help make the problem easier to manage, reducing errors and providing a clearer path to the solution.

Parentheses

In mathematics, parentheses \(( )\) are used to group part of an expression. They indicate that the operation inside should be performed first before others.
In the given exercise, the expression within the parentheses is \(18 - 6\).

• This tells you that subtraction needs to be calculated before addressing other operations outside of it.
• Once calculated, the result can be substituted back into the expression, simplifying the problem step by step.

It's crucial to notice that solving the inside of parentheses before any other operations ensures the correct order of operations is maintained.
Following this order not only simplifies your work but also ensures you arrive at the same correct answer every time.

Brackets

Brackets \([ ]\) are often used in mathematical expressions as an additional level of grouping, particularly when parentheses are already present.
In the problem, you find brackets encompassing \(12 - 6\). They serve as the next group to be simplified after the parentheses.

• Brackets provide clarity, showing the next operations to perform after those in parentheses.
• They help in structuring complex expressions, ensuring no part is overlooked.

Solving expressions with these structures systematically, as done here with simplifying \(12 - 6\) next, leads to upon reaching a simplified form of the expression \(7[6]\).
This makes handling mathematical problems orderly and less prone to mistakes.

Multiplication

After simplifying all the operations within parentheses and brackets, you then perform any remaining mathematical operations, such as multiplication.
Here, once simplified, the expression looks like \(7[6]\), which is essentially \(7 \times 6\).
To complete the task, simply calculate the multiplication:

• Multiply 7 by 6 to find the final value.
• This results in 42; thus, the original expression simplifies to this numerical value.

Completing multiplication at the end respects the order of operations, ensuring accuracy.
By understanding and applying each step, you can effectively tackle expressions of varying complexity.