Question

Evaluate each expression. $$ 10-[6-2 \cdot 2+(8-3)] \cdot 2 $$

Short answer

-4

Step-by-step solution

- Solve inside the innermost parentheses

Start by solving the operations inside the innermost parentheses first. Within the expression, identify the innermost parentheses: \[10-[6-2 \cdot 2+(8-3)] \cdot 2\]. Evaluate \(8-3\): \[10-[6-2 \cdot 2+5] \cdot 2\].

- Evaluate the multiplication inside the brackets

Next, perform the multiplication inside the brackets: \[6-2 \cdot 2+5\]. Evaluate \(2 \cdot 2\): \[6-4+5\].

- Perform the remaining operations inside the brackets

Finish evaluating the operations inside the brackets: \[6-4+5\]. First, subtract: \[6-4 = 2\]. Then, add: \[2+5 = 7\]. So, we now have: \[10-7 \cdot 2\].

- Evaluate the multiplication outside the brackets

Next, perform the multiplication outside the brackets: \[7 \cdot 2\]. Evaluate: \[7 \cdot 2 = 14\]. The expression is now: \[10-14\].

- Perform the final subtraction

Lastly, subtract \[14\] from \[10\]: \[10-14\]. This evaluates to: \[-4\].

Key concepts

PEMDAS

PEMDAS stands for Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). This is the standard order of operations used to solve mathematical expressions. Understanding this sequence is crucial when evaluating expressions as it ensures consistent and accurate results.

Here's a quick breakdown of the PEMDAS order:

• Parentheses: Solve expressions inside parentheses first.
• Exponents: Calculate powers or roots next.
• Multiplication and Division: From left to right.
• Addition and Subtraction: From left to right.

In the given exercise, applying PEMDAS helps in systematically breaking down the expression step by step.

Parentheses

Parentheses indicate that the operations inside them should be performed first. They are prioritized above other arithmetic operations. In the given exercise, parentheses play a vital role in structuring the order in which calculations are done.

Let's see how parentheses are applied in this problem:

The expression is: \10-[6-2 \cdot 2+(8-3)]\cdot 2\br>

• We start with the innermost parentheses: \(8-3\). This evaluates to \(5\).
• Replace \(8-3\) in the expression: \10-[6-2 \cdot 2+5]\cdot 2\br> This step ensures that operations inside parentheses are handled first before moving to others.

By handling the operations within parentheses first, it creates a structured approach to solving the rest of the expression.

Arithmetic Operations

Arithmetic operations consist of addition, subtraction, multiplication, and division. Each of these operations has a specific order when used together within an expression.

Here is how arithmetic operations were applied in the given exercise:

• First, we handled the subtraction and multiplication inside the parentheses: \(6-2 \cdot 2+5\).
• According to PEMDAS, perform multiplication first: \(2 \cdot 2 = 4\).
• The expression then becomes: \(6-4+5\).
• Perform the subtraction: \(6-4 = 2\).
• Then, add \(2+5\), which evaluates to \(7\).

Finally, we handle the multiplication and subtraction outside of the brackets as per the remaining order of operations:

• Multiplication: \(7\cdot 2 = 14\).
• Subtraction: \(10-14 = -4\).

Expression Evaluation

Expression evaluation is the process of breaking down and computing the value of a given mathematical expression. This process generally follows the order of operations guidelines, such as PEMDAS.

In the provided exercise, the evaluation follows these key steps:

• Identify and solve expressions within the innermost parentheses.
• Follow through with multiplication and addition/subtraction within the parentheses.
• Proceed with operations outside the parentheses, adhering to the same order of operations.

By strictly following these steps, the expression \(10-[6-2 \cdot 2+(8-3)]\cdot 2\) is evaluated step-by-step leading to the final result: