Order of Operations: BIDMAS / BODMAS

In mathematics, the order in which you perform calculations is critical. If you perform operations in the wrong order, you will get the wrong answer. We use the acronym BIDMAS to remember the hierarchy:

Brackets
\(()\), \([]\)
Indices
\(x^2\), \(\sqrt{x}\)
Division
\(\div\)
Multiplication
\(\times\)
Addition
\(+\)
Subtraction
\(-\)
Important Note on Priority: Division and Multiplication have equal priority and should be done from left to right. The same applies to Addition and Subtraction.
BIDMAS is sometimes referred to as BODMAS wih the 'O' meaning Of.

1. Complex Indices (Powers and Roots)

Indices include both powers and square roots. Square roots act like a bracket—you must evaluate the expression inside the root before taking the root itself.

Calculate: \(5 + \sqrt{20 + 5}\)
1. Inside the root: \(20 + 5 = 25\)
2. Take the root (Index): \(\sqrt{25} = 5\)
3. Addition: \(5 + 5 = 10\)

2. Hidden Brackets in Fractions

When you see a large fraction bar, there are implied brackets around the entire numerator (top) and the entire denominator (bottom).

Calculate: \(\frac{10 + 2 \times 5}{2^2}\)
This is treated as: \((10 + (2 \times 5)) \div (2^2)\)

Practice Questions

Try solving these on paper before checking the solutions below.

  1. \(12 - 3 \times 2 + 8\)
  2. \((5 + 3)^2 \div 4\)
  3. \(50 - [2 \times (10 - 3) + 4]\)
  4. \(\frac{\sqrt{16} + 2^3}{2 \times 3}\)
  5. \(4 \times (2 + 3^2) - \frac{10}{2}\)

Detailed Solutions

1. Solution: \(12 - 3 \times 2 + 8\)
Multiplication first: \(3 \times 2 = 6\) Equation is now: \(12 - 6 + 8\) Left to Right: \(12 - 6 = 6\), then \(6 + 8 = 14\)
Answer: 14
2. Solution: \((5 + 3)^2 \div 4\)
Brackets first: \(5 + 3 = 8\) Indices (Power): \(8^2 = 64\) Division: \(64 \div 4 = 16\)
Answer: 16
3. Solution: \(50 - [2 \times (10 - 3) + 4]\)
Inner Bracket: \(10 - 3 = 7\) Outer Bracket (Multiplication): \(2 \times 7 = 14\) Outer Bracket (Addition): \(14 + 4 = 18\) Final Subtraction: \(50 - 18 = 32\)
Answer: 32
4. Solution: \(\frac{\sqrt{16} + 2^3}{2 \times 3}\)
Top Indices: \(\sqrt{16} = 4\) and \(2^3 = 8\) Top Addition: \(4 + 8 = 12\) Bottom Multiplication: \(2 \times 3 = 6\) Final Division: \(12 \div 6 = 2\)
Answer: 2
5. Solution: \(4 \times (2 + 3^2) - \frac{10}{2}\)
Inside Bracket (Index): \(3^2 = 9\) Inside Bracket (Addition): \(2 + 9 = 11\) Multiplication: \(4 \times 11 = 44\) Fraction/Division: \(10 \div 2 = 5\) Subtraction: \(44 - 5 = 39\)
Answer: 39