Problem 56 - Fundamentals of Mathematics

Question

Find the value of \(-3(2-6)-12\).

Step-by-Step Solution

Answer

The value of the expression is 0.

Step 1: Solve Inside the Parentheses

Examine the expression inside the parentheses: \[ 2 - 6. \]Perform the subtraction inside the parentheses:\[ 2 - 6 = -4. \]Now the expression becomes:\[ -3(-4) - 12. \]

Step 2: Multiply the Numbers

Now, multiply the number outside the parentheses by the result from the parentheses:\[ -3 \times (-4) = 12. \]Now the expression simplifies to:\[ 12 - 12. \]

Step 3: Perform the Final Subtraction

Subtract the numbers in the expression:\[ 12 - 12 = 0. \]This leaves us with the final result.

Key Concepts

Parentheses

In mathematics, parentheses are used to indicate that the enclosed operations should be performed first in an expression. They play a crucial role in managing the order of operations, which is sometimes abbreviated as PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction).
When you see parentheses, it's your cue to tackle whatever is inside them before moving on to other operations. This ensures calculations are accurate and follow a universal rule.

• Identify the expression within the parentheses.
• Perform necessary arithmetic operations inside first.
• In the given exercise, the operation inside the parentheses was a subtraction: \[ 2 - 6 = -4 \].

Handling operations inside parentheses first lays down the groundwork for the rest of your calculations.

Subtraction

Subtraction is one of the four fundamental arithmetic operations and involves taking one number away from another. It is essentially the reverse of addition. In the order of operations, subtraction is typically performed after any operations enclosed in parentheses, multiplication, and division.
Understanding subtraction is crucial when simplifying expressions, as it can significantly change the outcome.

• Identify the numbers involved in the subtraction.
• Calculate the difference between these numbers.
• Example: In the exercise, the subtraction inside the parentheses was solved as \( 2 - 6 = -4 \).

Subtraction appears again in the final step of the exercise, where \(12 - 12 \) gives 0. Both instances demonstrate the importance of carrying out subtraction carefully to achieve the correct answer.

Multiplication

Multiplication is another key arithmetic operation that denotes repeated addition. It is often represented by the symbol \( \times \) or by placing numbers next to each other, such as in \(-3(-4)\). In order of operations, it generally follows parentheses operations.In the context of the order of operations, multiplication must be performed before addition or subtraction unless dictated by parentheses.

• Identify which numbers need to be multiplied.
• Apply multiplication rules such as the sign rule: the product of two negative numbers is positive.
• In the exercise, multiplying \(-3\) by the result \(-4\) inside the parentheses gives \(12\).

Multiplication is a pivotal step that can lead to a correct or incorrect calculation if not done properly, as it can flip the sign or the entire result of the expression.