GCSE Maths Practice: order-of-operations-bidmas

Question 10 of 10

This Higher-tier question tests your ability to apply BIDMAS to nested brackets, combining powers, multiplication, and subtraction accurately.

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\( \begin{array}{l}\text{Work out }8 + (3 \times (2^2 + 1)) - 5\\\text{ using BIDMAS.}\end{array} \)

Choose one option:

Always start from the innermost bracket and move outward. Write each step clearly to avoid skipping operations.

Mastering Nested Brackets with BIDMAS

This higher-tier question explores BIDMAS through nested brackets—an essential skill for GCSE students preparing for algebraic and problem-solving questions. BIDMAS ensures that every calculation follows the correct order: Brackets, Indices, Division, Multiplication, Addition, Subtraction.

Step-by-Step Method

  1. Innermost brackets: Start with what’s inside the deepest bracket, including any powers.
  2. Next level brackets: Simplify the remaining bracket by performing multiplication or division.
  3. Outer operations: Finally, complete addition or subtraction.

Following these steps prevents common errors that occur when expressions have more than one set of brackets or powers.

Worked Examples

Example 1: \(8 + (3 \times (2^2 + 1)) - 5\).
Innermost bracket: \(2^2 + 1 = 5\).
Then \(3 \times 5 = 15\).
Finally \(8 + 15 - 5 = 18\).

Example 2: \(6 + (4 \times (3^2 - 2)) - 8\).
Power: \(3^2 = 9\).
Inner bracket: \(9 - 2 = 7\).
Multiply: \(4 \times 7 = 28\).
Outer steps: \(6 + 28 - 8 = 26\).

Example 3: \(5 + (2 \times (4^2 - 3)) - 1\).
Powers first: \(4^2 = 16\).
Inside bracket: \(16 - 3 = 13\).
Multiply: \(2 \times 13 = 26\).
Final step: \(5 + 26 - 1 = 30\).

Common Mistakes

  • Solving brackets from the outside instead of inside first.
  • Skipping powers or calculating them after multiplication.
  • Not keeping track of subtraction signs outside brackets.
  • Forgetting that multiplication affects the entire inner result.

Understanding the Concept

Nested brackets show up frequently in algebra, geometry, and even coding. In algebraic equations, each bracket may represent a function or operation that depends on the result of another. Learning to process them in order builds the foundation for expanding and simplifying expressions later.

Real-Life Application

Nested operations appear in formulas for compound interest, scientific equations, and spreadsheet functions. For example, an engineer might calculate force as \(F = (m \times (a + g)) - r\), where each variable and bracket changes the result. Following BIDMAS guarantees consistent outcomes whether you’re programming, calculating physics problems, or using Excel formulas.

FAQ

Q1: What does ‘nested’ mean?
A: Brackets within brackets. Always solve the inner ones first.

Q2: Are powers always done before brackets?
A: No—evaluate powers inside brackets before multiplying or dividing, but complete the brackets before moving on to outside operations.

Q3: What if there are multiple brackets at the same level?
A: Solve each separately, left to right, before handling the outer operations.

Exam Tip

Write each calculation on a new line, especially when more than one set of brackets is involved. Label each stage (e.g., ‘inner bracket done,’ ‘power done’) to track your logic. This reduces errors and demonstrates clear working—earning method marks even if your final answer slips.

Summary

Nested brackets challenge your understanding of BIDMAS beyond simple arithmetic. Always work from the deepest point outward, evaluate powers before multiplying, and finish with addition or subtraction. With careful, step-by-step reasoning, you can confidently handle even the most complex multi-layered problems on the GCSE Higher paper.