Question

Evaluate each expression. $$ 6-[3 \cdot 5+2 \cdot(3-2)] $$

Short answer

-11

Step-by-step solution

Evaluate the inner parentheses

First, solve the expression inside the inner parentheses: \( 3 - 2 \). So, we have: \( 3 - 2 = 1 \)

Simplify the terms inside the brackets

Next, substitute the result from Step 1 into the expression and perform the multiplications inside the brackets: \[ 3 \times 5 + 2 \times (1) \]. This simplifies to: \[ 3 \times 5 + 2 \times 1 = 15 + 2 = 17 \]

Perform the final subtraction

Finally, subtract the result from Step 2 from 6: \( 6 - 17 \). So, the final answer is: \( 6 - 17 = -11 \)

Key concepts

Evaluating Expressions

Evaluating expressions means finding the value of an expression by following the correct order of operations. In mathematics, the order of operations is a set of rules that dictates the sequence in which operations should be performed to ensure consistent results. These rules are often remembered using the acronym PEMDAS:

• P: Parentheses first
• E: Exponents (i.e., powers and roots, etc.)
• MD: Multiplication and Division (from left to right)
• AS: Addition and Subtraction (from left to right)

The given expression to evaluate here is: \( 6-[3 \times 5+2 \times (3-2)] \). To evaluate this expression, you must carefully follow each step of the order of operations. This ensures that you correctly combine all the numbers and operations to find the final value.

Parentheses

Parentheses are used in mathematical expressions to indicate which operations should be performed first. When you see parentheses, you should always start by evaluating the expression inside them. This prioritization prevents errors in calculations, especially when multiple operations are involved.

In our given expression: \( 6-[3 \times 5 + 2 \times (3-2)] \). The first step is to solve \( 3 - 2 \) inside the parentheses. Here, \( 3 - 2 = 1 \). If there are more parentheses within brackets, solve the innermost ones first, working your way out. This careful approach helps manage complex expressions with multiple layers and prevents miscalculations.

Subtraction

Subtraction is one of the basic arithmetic operations. It represents the operation of removing objects from a collection. In mathematical expressions, you perform subtraction after handling the operations inside parentheses and after completing any multiplications or divisions.

For the given expression \( 6 - [3 \times 5 + 2 \times (3-2)] \), after evaluating everything inside the brackets, you simplify it first. Substitute the simplified value to get \( 6 - 17 \). Finally, perform the subtraction: \( 6 - 17 = -11 \). Remember to always subtract by following the order as per PEMDAS rules to avoid mistakes.