How to calculate squares and cubes mentally

Squaring two-digit numbers in your head looks like a party trick, but it's really just one small piece of algebra applied consistently. Once the method clicks, numbers like 47² or 63² stop being calculator problems.

The core trick: use a nearby round number

The method relies on this identity:

Rearranged, that means:

In plain terms: instead of squaring a number directly, move to a nearby round number, do an easier multiplication, then correct for the difference.

Worked example: 47²

47 is close to 50. So we set b = 3 (the gap between 47 and 50).

• 47 + 3 = 50
• 47 − 3 = 44
• 50 × 44 = 2200 (easy — just 44 × 5, then add a zero)
• Add b² = 3² = 9

Worked example: 63²

63 is close to 60. Set b = 3.

• 63 + 3 = 66... this isn't round, so instead go the other way: use 60 as the base directly.

Simpler approach for numbers ending near a multiple of 10 — split into tens and units:

This is the standard (a+b)² = a² + 2ab + b² expansion, and it's often the more reliable method to practice first since it works the same way every time.

A simpler rule for numbers close to 50

If a number is 50 + n, its square follows a clean pattern:

Example: 54² → n = 4 → 2500 + 400 + 16 = 2916.

Cubes: build on the square

Once you're fast at squares, cubes just need one more multiplication: a³ = a² × a. The time-saver is having the square memorized or computed quickly, so you're only doing one more multiplication step, not three.

Where this actually helps

Bank exam and competitive exam quant sections frequently need squares/cubes for approximation, simplification questions, and number series. Being able to compute 47² in under 5 seconds — instead of reaching for long multiplication — adds up across a timed paper.