Class 8 CBSE Maths: Cubes, Coordinate Geometry, Factorisation & Identities: quick revision, 18 bites

A Perfect Cube Needs Every Prime in Triples

Cubes hide in prime factor triples.

A number is a **perfect cube** only when every prime factor appears a **multiple of 3 times** in its prime factorisation.

Takeaway. To test or build a cube, count prime exponents in groups of three.

Abscissa vs Ordinate — Name the Coordinates

Two numbers, two names — don't mix them.

Takeaway. In (x, y), x is the abscissa and y is the ordinate.

The Four Identities That Solve Half the Paper

Four identities, endless questions.

Takeaway. Recognising the pattern is faster than expanding every term.

Find the Smallest Multiplier to Make a Cube

One missing prime ruins the cube.

Takeaway. The multiplier is the product of primes whose exponents fall short of a multiple of 3.

Find the Smallest Divisor to Make a Cube

Divide away the leftovers.

Takeaway. Dividing removes surplus primes; multiplying fills missing ones.

Factorisation Step 1: Pull Out the HCF

Always hunt the common factor first.

Takeaway. HCF-outside, quotient-inside — the universal first move in factorisation.

Sign Flip Trap: (3 − a) vs (a − 3)

Flip the sign, save the factor.

Takeaway. Whenever you see (a − b) and (b − a), flip one sign and factor.

Squaring Big Numbers Without Long Multiplication

197² in one line — no calculator.

Takeaway. Round the number, apply the identity, adjust back.

Difference of Squares: The Fastest Identity

Big squares cancel in one step.

Takeaway. Spot a² − b², split into sum × difference.

Continued Products Collapse in Pairs

Multiply in pairs, not all at once.

When you see $(3x-2y)(3x+2y)(9x^2+4y^2)$, pair the conjugates first — they collapse into a difference of squares.

Takeaway. Conjugate pairs always multiply into a difference of squares.

The Three-Variable Bridge: x² + y² from x − y and xy

Two facts, one missing square.

Takeaway. Whenever a question gives (x ± y) and xy, use the squared identity.

Recognise a Hidden Perfect Square

That ugly expression is a square.

Expressions like $64x^2 + 144xy + 81y^2$ look messy but are actually $(8x + 9y)^2$ in disguise.

Takeaway. Spot the pattern, substitute once, done.

Plot Points and Test for Collinearity

Three points, one line — check the slope.

Takeaway. Equal slopes between consecutive points = collinear.

Rhombus Diagonals Bisect Each Other

Find the fourth vertex with one midpoint.

In a rhombus $ABCD$, the diagonals $AC$ and $BD$ share the same midpoint — use it to find a missing vertex.

Takeaway. Midpoint of one diagonal = midpoint of the other in any parallelogram.

Show Prime Factorisation — Marks Live There

Skip the working, lose the marks.

Takeaway. Examiners award method marks — never jump to the answer.

Odd Cube, Odd Root — Even Cube, Even Root

Parity survives cubing.

Takeaway. Cubing never changes whether a number is odd or even.

Cubes in Real Life: Water Tanks and Dice

Cubes aren't just textbook shapes.

Takeaway. Cube roots turn volume back into side length — a real engineering move.

Don't Expand — Factor First

Expanding wastes time and invites errors.

Takeaway. Identity-spotting is faster and safer than brute-force expansion.