How to Solve Expressions with Grouping Symbols

Solving expressions with grouping symbols like parentheses, brackets, and braces can seem tricky at first, but once you understand the order of operations, it becomes much easier.

Order of Operations

To solve these expressions, we follow the PEMDAS/BODMAS rule:

• P/B: Parentheses/Brackets
• E/O: Exponents/Orders (like squares and square roots)
• M/D: Multiplication and Division (left to right)
• A/S: Addition and Subtraction (left to right)

Step-by-Step Example

Let’s work through an example to see how this rule applies. Consider the expression:
$3 + (2 times (4 + 5))$

1. Solve Inside the Innermost Parentheses
   First, solve the expression inside the innermost parentheses:
   $4 + 5 = 9$

   Now, the expression looks like this:
   $3 + (2 times 9)$

2. Perform Multiplication and Division
   Next, perform the multiplication:
   $2 times 9 = 18$

   Now, the expression is:
   $3 + 18$

3. Perform Addition and Subtraction
   Finally, perform the addition:
   $3 + 18 = 21$

   So, the value of the expression $3 + (2 times (4 + 5))$ is 21.

More Complex Example

Let’s try a more complex example:
$5 times [2 + (3 – 1)^2]$

1. Solve Inside the Innermost Parentheses
   First, solve the expression inside the innermost parentheses:
   $3 – 1 = 2$

   Now, the expression looks like this:
   $5 times [2 + 2^2]$

2. Solve the Exponents
   Next, solve the exponent:
   $2^2 = 4$

   Now, the expression is:
   $5 times [2 + 4]$

3. Solve Inside the Brackets
   Now, solve the expression inside the brackets:
   $2 + 4 = 6$

   Now, the expression is:
   $5 times 6$

4. Perform Multiplication
   Finally, perform the multiplication:
   $5 times 6 = 30$

   So, the value of the expression $5 times [2 + (3 – 1)^2]$ is 30.

Tips for Success

1. Work from the inside out: Always start with the innermost grouping symbols and work your way out.
2. Follow the order of operations: Stick to PEMDAS/BODMAS to avoid mistakes.
3. Double-check your work: It’s easy to make small errors, so always go back and verify each step.

By understanding and following these steps, solving expressions with grouping symbols becomes a straightforward process.