Question

Evaluate each expression. $$ 1-(4 \cdot 3-2+2) $$

Short answer

-11

Step-by-step solution

Evaluate the expression inside the parentheses

First, evaluate the expression inside the parentheses: \( (4 \times 3 - 2 + 2) \).

Perform the multiplication

Calculate the multiplication: \( 4 \times 3 = 12 \). Now the expression inside the parentheses becomes \( (12 - 2 + 2) \).

Perform the subtraction

Next, perform the subtraction: \( 12 - 2 = 10 \). So the expression is now \( (10 + 2) \).

Perform the addition

Now, add the remaining numbers: \( 10 + 2 = 12 \). Hence, the expression inside the parentheses evaluates to 12.

Complete the outer expression

Finally, use the result from the parentheses to complete the outer expression: \( 1 - 12 \). Therefore, \( 1 - 12 = -11 \).

Key concepts

Order of Operations

When solving mathematical expressions, it's crucial to follow a specific sequence known as the 'Order of Operations.' This ensures everyone evaluates expressions the same way and gets the same result. The order is often remembered using the acronym PEMDAS, which stands for:

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

Let's apply this to our example.
Applying PEMDAS, we first evaluate the parentheses, followed by multiplication, subtraction and addition, and finally, handle any remaining operations outside the parentheses.

Parentheses

Parentheses are used to group parts of an expression that need to be evaluated first. In math, anything inside parentheses is given top priority.
In our example, the expression inside parentheses is \( 4 \times 3 - 2 + 2 \).
To solve it, start by handling the operations inside the parentheses step-by-step:

• First, perform the multiplication: \( 4 \times 3 = 12 \).
• Next, perform the subtraction: \( 12 - 2 = 10 \).
• Finally, add the remaining number: \( 10 + 2 = 12 \).

Thus, what's inside the parentheses simplifies to 12. Now we replace the original parentheses expression with its result, 12. The expression becomes \( 1 - 12 \).

Arithmetic Operations

After simplifying what's inside the parentheses, the next step is to carry out any remaining arithmetic operations. These include addition, subtraction, multiplication, and division.
In our given expression, after solving the parentheses, we are left with a simple arithmetic problem: \( 1 - 12 \).
Let's break it down:

• Subtraction: Subtraction is one of the basic arithmetic operations where you remove a number (the subtrahend) from another number (the minuend). Here, 12 is subtracted from 1. This means we are finding how much less 1 is compared to 12.

So, \( 1 - 12 = -11 \). That's the final answer. Understanding these operations helps in breaking down and solving complex mathematical expressions efficiently.