Evaluate the Mathematical Expression Using BODMAS/PEMDAS

To evaluate the given expression, we need to follow the order of operations, often remembered by acronyms like BODMAS or PEMDAS.

• Brackets (or Parentheses)
• Orders (or Exponents, roots)
• Division and Multiplication (from left to right)
• Addition and Subtraction (from left to right)

The given expression is:

\((13 \times 8 - \sqrt{81} \times 6 + 1 ) \div (\sqrt{49} + 72 - 18 \times 2 + 19)\)

We will evaluate the numerator and the denominator separately.

Evaluating the Numerator: \(13 \times 8 - \sqrt{81} \times 6 + 1\)

First, calculate the square root:

• \(\sqrt{81} = 9\)

Substitute this value back into the numerator expression:

\(13 \times 8 - 9 \times 6 + 1\)

Next, perform the multiplications from left to right:

• \(13 \times 8 = 104\)
• \(9 \times 6 = 54\)

Substitute these values back:

\(104 - 54 + 1\)

Finally, perform addition and subtraction from left to right:

• \(104 - 54 = 50\)
• \(50 + 1 = 51\)

So, the value of the numerator is 51.

Evaluating the Denominator: \(\sqrt{49} + 72 - 18 \times 2 + 19\)

First, calculate the square root:

• \(\sqrt{49} = 7\)

Substitute this value back into the denominator expression:

\(7 + 72 - 18 \times 2 + 19\)

Next, perform the multiplication:

• \(18 \times 2 = 36\)

Substitute this value back:

\(7 + 72 - 36 + 19\)

Finally, perform addition and subtraction from left to right:

• \(7 + 72 = 79\)
• \(79 - 36 = 43\)
• \(43 + 19 = 62\)

So, the value of the denominator is 62.

Performing the Final Division

Now, divide the numerator by the denominator:

\(\frac{\text{Numerator}}{\text{Denominator}} = \frac{51}{62}\)

The result of the evaluation is \(\frac{51}{62}\).

Revision Table: Order of Operations (BODMAS/PEMDAS)

Additional Information: Understanding Arithmetic Operations and Square Roots

This problem combines basic arithmetic operations (addition, subtraction, multiplication, division) with square roots.

• Arithmetic Operations: These are the four fundamental ways to combine numbers. Following the correct order is crucial for getting the right answer in complex expressions.
• Square Root: The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 81 (\(\sqrt{81}\)) is 9 because \(9 \times 9 = 81\). Similarly, the square root of 49 (\(\sqrt{49}\)) is 7 because \(7 \times 7 = 49\).
• Expressions: A mathematical expression is a combination of numbers, variables, and operation symbols. Evaluating an expression means finding its numerical value.

Always break down complex expressions into smaller, manageable steps following the BODMAS/PEMDAS rule to avoid errors.