Problem 49 - Introductory Algebra for College Students

Question

In Exercises \(29-72,\) use the order of operations to simplify each expression. $$2[5+2(9-4)]$$

Step-by-Step Solution

Answer

The simplified expression for \( 2[5+2(9-4)] \) is 30.

Step 1: Calculation within Parentheses

First, calculate the operation within the parentheses \( (9-4) \), which results in 5.

Step 2: Calculation within Brackets

Next, within the brackets, perform the operation \( 5+2*5 \) (obtained from step 1). According to BIDMAS/BODMAS/PEDMAS, multiplication should be done first, which gives \( 5+10 \), and then addition, which gives 15.

Step 3: Final Calculation

Finally, multiply the number outside of the brackets by the outcome from step 2, which gives \( 2*15=30 \)

Key Concepts

Understanding Parentheses

In mathematics, parentheses are used to indicate that the operations within them should be performed as a priority. Think of them as a way to group numbers and operations together. When you see parentheses in an expression, you always handle the operations inside them first.
In the original exercise, we have \( (9-4) \). Here, we perform the subtraction first, which yields 5.
Using parentheses helps to clarify expressions, making sure there are no misunderstandings about which operations to execute first.

• The contents inside parentheses can be any kind of mathematical operation, including addition, subtraction, multiplication, or division.
• Once the operations inside parentheses are simplified, move on to the next priority operation outside the parentheses as per the order of operations.

The Role of Multiplication in Order of Operations

Once the expression inside the parentheses is simplified, multiplication often comes next in the order of operations rules, also known as PEDMAS or BIDMAS. Multiplication must be executed before addition and subtraction unless otherwise indicated by parentheses.
In the given exercise, after simplifying \((9-4)\) into 5, the expression becomes \(5 + 2 \times 5\). The multiplication is performed first, turning the expression into \( 5 + 10 \).

• This rule prevents errors in calculating expressions and ensures consistency in mathematics.
• If multiplication were considered after addition, the results would be different and incorrect.

Hence, understanding the placement and role of multiplication in expressions helps to solve mathematical problems accurately.

Executing Addition Correctly

In the order of operations, addition is typically one of the last tasks performed, after parentheses, exponents, multiplication, and division have been handled.
The exercise simplifies \( 5 + 2 \times 5 \) by calculating the multiplication first (giving us 10), and then proceeds to addition to yield \( 5 + 10 = 15 \).

• Addition combines numbers to give a total or sum.
• While it is straightforward, ensuring addition is performed in the right order is crucial for solving problems correctly.

By consistently following the order of operations, we avoid mistakes and ensure that the addition produces the correct and expected result.