Understanding the Problem: Simplifying the Expression

The question asks us to simplify a given mathematical expression involving multiple operations and nested brackets. To solve this type of problem accurately, we need to follow the correct order of operations. A widely used rule for the order of operations is BODMAS or PEMDAS.

What is the Order of Operations (BODMAS/PEMDAS)?

The BODMAS rule helps us remember the sequence in which operations should be performed:

• Brackets first (Parentheses) - Solve the expressions inside brackets: (), {}, []. Work from the innermost brackets outwards.
• Orders (Exponents) - Calculate powers and square roots.
• Division and Multiplication - Perform division and multiplication from left to right.
• Addition and Subtraction - Perform addition and subtraction from left to right.

Applying BODMAS to Simplify the Expression

The given expression is: \(2.5 \times [144 \div 198 \times \{121 \times 81 \div (11 \times 9)\}]\)

Let's simplify step-by-step following the BODMAS rule:

Step 1: Innermost Brackets (Parentheses)

First, calculate the expression inside the innermost parentheses \((11 \times 9)\):

\((11 \times 9) = 99\)

The expression becomes: \(2.5 \times [144 \div 198 \times \{121 \times 81 \div 99\}]\)

Step 2: Curly Brackets

Next, calculate the expression inside the curly brackets \(\{121 \times 81 \div 99\}\). Within these brackets, we have multiplication and division. We perform them from left to right.

\(121 \times 81 = 9801\)

Now, perform the division:

\(9801 \div 99\)

To make this division easier, we can factorize the numbers or perform long division. Note that \(121 = 11^2\), \(81 = 9^2\), and \(99 = 9 \times 11\). So, \( \frac{121 \times 81}{99} = \frac{11 \times 11 \times 9 \times 9}{11 \times 9} \). Cancelling out \(11 \times 9\) from numerator and denominator, we get \(11 \times 9 = 99\). Alternatively, performing division \(9801 \div 99\) gives 99.

So, \(\{121 \times 81 \div 99\} = 99\)

The expression becomes: \(2.5 \times [144 \div 198 \times 99]\)

Step 3: Square Brackets

Now, calculate the expression inside the square brackets \([144 \div 198 \times 99]\). Within these brackets, we have division and multiplication. We perform them from left to right.

First, perform the division:

\(144 \div 198\)

This can be written as a fraction: \(\frac{144}{198}\). Both numbers are divisible by 18 (since 144 = 18 * 8 and 198 = 18 * 11). \(\frac{144 \div 18}{198 \div 18} = \frac{8}{11}\)

So, \(144 \div 198 = \frac{8}{11}\).

Now, perform the multiplication:

\(\frac{8}{11} \times 99\)

\(\frac{8}{\cancel{11}} \times \cancel{99}^{9} = 8 \times 9 = 72\)

So, \([144 \div 198 \times 99] = 72\)

The expression becomes: \(2.5 \times 72\)

Step 4: Final Multiplication

Finally, perform the multiplication outside the brackets:

\(2.5 \times 72\)

We can write \(2.5\) as \(\frac{5}{2}\). \(\frac{5}{2} \times 72 = 5 \times \frac{72}{2} = 5 \times 36\)

\(5 \times 36 = 180\)

So, the simplified value of the expression is 180.

The final simplified value of the expression \(2.5 \times [144 \div 198 \times \{121 \times 81 \div (11 \times 9)\}]\) is 180.

Revision Table: Order of Operations

Additional Information: Simplifying Complex Expressions

When simplifying complex mathematical expressions, careful attention to the order of operations is crucial. A single mistake in the sequence of operations can lead to an incorrect result. For expressions with multiple levels of brackets, always start with the innermost set of brackets and work your way outwards. Remember that division and multiplication have equal priority and should be performed from left to right as they appear in the expression. Similarly, addition and subtraction have equal priority and are also performed from left to right.

Practice with various examples helps build confidence in applying the BODMAS/PEMDAS rule accurately to simplify mathematical expressions.