GCSE Maths Practice: integers-and-directed-numbers

Question 8 of 10

This advanced BIDMAS problem combines powers, multiplication, and subtraction of negative numbers.

‹ ›

\( \begin{array}{l}\text{Evaluate } 5 \times (-4) - [(-3)^2 \times 2].\end{array} \)

Choose one option:

Always handle indices before multiplication and subtraction, especially with negatives inside brackets.

Mastering BIDMAS with Powers and Negatives

This question takes multi-step integer arithmetic to the next level by adding an index (square) to one of the negative terms. Powers of negative numbers introduce another layer of logic to BIDMAS. The key principle: when a negative number is raised to a power, the result depends on whether the power is even or odd.

Power and Sign Rules

  • If the power is even, the result is positive (e.g., (−3)2 = +9).
  • If the power is odd, the result stays negative (e.g., (−3)3 = −27).

It’s also crucial to note that (−3)2 is not the same as −32. The first has brackets and squares the whole negative number, while the second squares only the 3 and leaves the negative sign outside.

Breaking Down the Expression

Consider the upgraded expression: 5 × (−4) − [ (−3)2 × 2 ].

  1. Handle the power first: (−3)2 = 9.
  2. Do both multiplications: 5 × (−4) = −20, and 9 × 2 = 18.
  3. Subtract correctly: −20 − 18 = −38.

If the question had been 5 × (−4) − [ (−3) × 2 ], the result would instead be −14. Notice how introducing the power dramatically changes the outcome.

Common Traps

  • Forgetting to square the negative when brackets are present.
  • Squaring without brackets (−3^2 = −9, not +9).
  • Adding before performing multiplication or powers.
  • Assuming subtracting a negative always means addition—only true if that negative sits directly after a subtraction sign.

Worked Examples (Different from This Question)

  • Example 1: (−2)2 × 3 − (−5) = 4×3−(−5) = 12 + 5 = 17.
  • Example 2: 7 − [ (−4)2 × 2 ] = 7 − 32 = −25.
  • Example 3: (−6) × 2 − (−3)2 = −12 − 9 = −21.

Reasoning Strategy

Follow a strict order: powers → multiplication → addition/subtraction. When brackets are involved, resolve the deepest ones first. Keep careful track of negatives. Many Higher-tier exam questions deliberately include multiple minus signs or brackets to test whether students apply the correct sequence.

Real-World Relevance

Powers and negatives appear in finance (percentage decreases), physics (directional quantities squared in energy equations), and computer science (signed integer operations). Understanding how they interact ensures correct reasoning across disciplines.

FAQs

Q1: Why does an even power make a negative positive?
A: Because a number multiplied by itself an even number of times cancels the direction each pair of negatives introduces.

Q2: How can I tell whether to keep brackets?

A: Use brackets whenever the power applies to the entire negative number. Without them, only the value part is squared, not the sign.

Q3: How do I avoid mistakes in longer expressions?

A: Rewrite each stage clearly—mark operations you’ve done and carry the sign carefully into the next step.

Study Tip

Before squaring or cubing negatives, rewrite the expression with brackets to prevent confusion. Then apply BIDMAS methodically: powers first, then multiplications, then additions/subtractions. Consistent structure is what distinguishes top-grade students from those who lose marks on small sign errors.