Simplification

medium importance~2 Q in Tier 126 formulas⚡ 14 shortcuts5 subtopics

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Subtopic 2 of 5·← BODMAS, 'of', brackets & vinculum·Surds & indices →

Algebraic identities in numerical simplification

Many CGL simplification questions are identities in disguise. Spot the shape, name $a$ and $b$ , and the answer is often just $a+b$ or $a-b$ .

Looks like	Equals
$\frac{a^3 + b^3}{a^2 - ab + b^2}$	$a + b$
$\frac{a^3 - b^3}{a^2 + ab + b^2}$	$a - b$
$\frac{a^2 - b^2}{a - b}$	$a + b$
$\frac{(a+b)^2 - (a-b)^2}{ab}$	4
$\frac{(a+b)^2 + (a-b)^2}{a^2 + b^2}$	2

Also: if $a + b + c = 0$ then $a^3 + b^3 + c^3 = 3abc$ .

📖 Notes 🧩 5 question types ⚡ Formulas 🎯 12 practice Q

Detailed notes

Why identities?

Many simplification questions are just algebraic identities with numbers pasted in. If you can see the shape, the answer comes in one line — no cubing of 3.3 needed.

The identities you must know

Identity	Read as
$(a+b)^2 = a^2 + 2ab + b^2$	square of sum
$(a-b)^2 = a^2 - 2ab + b^2$	square of difference
$(a+b)(a-b) = a^2 - b^2$	difference of squares
$a^3 + b^3 = (a+b)(a^2 - ab + b^2)$	sum of cubes
$a^3 - b^3 = (a-b)(a^2 + ab + b^2)$	difference of cubes
$(a \pm b)^3 = a^3 \pm b^3 \pm 3ab(a \pm b)$	cube of sum/difference
$a^3 + b^3 + c^3 - 3abc = (a+b+c)(a^2+b^2+c^2-ab-bc-ca)$	three-variable

Zero-sum magic: if $a + b + c = 0$ , the long factor is not zero, so $a^3 + b^3 + c^3 = 3abc$ . Example: $25^3 + (-17)^3 + (-8)^3 = 3 \times 25 \times (-17) \times (-8) = 10200$ .

Identity fractions (the most asked shape)

Because $a^3 + b^3 = (a+b)(a^2 - ab + b^2)$ , the bracket cancels: $\frac{a^3 + b^3}{a^2 - ab + b^2} = a+b, \qquad \frac{a^3 - b^3}{a^2 + ab + b^2} = a-b$ Example: $\frac{(15.2)^3 - (5.2)^3}{(15.2)^2 + 15.2 \times 5.2 + (5.2)^2} = 15.2 - 5.2 = 10$ . Check the middle sign: minus in the denominator pairs with plus on top.

Shortcuts from the squares

$(a+b)^2 + (a-b)^2 = 2(a^2 + b^2)$
$(a+b)^2 - (a-b)^2 = 4ab$
$a^2 + b^2 = (a+b)^2 - 2ab$
$a^3 + b^3 = (a+b)^3 - 3ab(a+b)$ Example: with $a+b = 7$ , $ab = 12$ : $a^3 + b^3 = 343 - 3 \times 12 \times 7 = 91$ .

The x + 1/x family (asked again and again)

If $x + \frac{1}{x} = k$ , then:

$x^2 + \frac{1}{x^2} = k^2 - 2$
$x^3 + \frac{1}{x^3} = k^3 - 3k$
$x^4 + \frac{1}{x^4} = (k^2-2)^2 - 2$ If $x - \frac{1}{x} = k$ , then $x^2 + \frac{1}{x^2} = k^2 + 2$ and $x^3 - \frac{1}{x^3} = k^3 + 3k$ . Example: $x + \frac{1}{x} = 4$ → $x^2 + \frac{1}{x^2} = 16 - 2 = 14$ .

Numbers near a round value

$105^2 - 95^2 = (105-95)(105+95) = 10 \times 200 = 2000$ . Use $a^2 - b^2$ whenever two squares sit close together.

Cubes when a sum and a product are given

If $a+b$ and $ab$ are known, never cube anything big:

$a^2 + b^2 = (a+b)^2 - 2ab$
$a^3 + b^3 = (a+b)^3 - 3ab(a+b)$ Example: $a+b = 10$ , $ab = 21$ → $a^2 + b^2 = 100 - 42 = 58$ . The same two facts give $a^3 - b^3$ if you first find $a-b$ from $(a-b)^2 = (a+b)^2 - 4ab$ .

How to attack any identity question

Look at the denominator first (if there is one) — it tells you which factor will cancel.
Match the signs to pick plus or minus identities; a wrong middle sign is the classic trap.
Substitute only at the last step; keep $a$ , $b$ symbolic until the expression is fully factored.

Quick revision

$\frac{a^3 \pm b^3}{a^2 \mp ab + b^2} = a \pm b$ .
$a+b+c = 0$ → cubes sum to $3abc$ .
$x+\frac1x = k$ : squares give $k^2-2$ , cubes give $k^3-3k$ .
$(a+b)^2 - (a-b)^2 = 4ab$ .
Difference of squares for near numbers.

Types of questions asked

Every way this subtopic shows up in exams — how to recognise it, the formula or logic to use, and a solved example.

Type 1: Identity fraction: cubes over the quadratic factorvery common2 practice Q

How to spot it:

A fraction with a³ + b³ (or a³ − b³) on top — often decimals — and a² − ab + b² (or a² + ab + b²) below.

\frac{a^3 \pm b^3}{a^2 \mp ab + b^2} = a \pm b

Check the sign in the denominator: minus below pairs with plus above; plus below pairs with minus above.
Cancel the common quadratic factor.
The answer is just a + b (or a − b).

Why: $a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2)$ — the denominator is literally the second factor.

Example: $\frac{(15.2)^3 - (5.2)^3}{(15.2)^2 + 15.2 \times 5.2 + (5.2)^2}$ = ?

Denominator is $a^2+ab+b^2$ → value $= 15.2 - 5.2 = 10$ .

Type 2: Squares near each other; (x+y)² and (x−y)² combosvery common2 practice Q

How to spot it:

$105^2 - 95^2$ -type questions, or the value of $(x+y)^2 + (x-y)^2$ for given x, y.

a^2 - b^2 = (a-b)(a+b),\quad (x+y)^2 + (x-y)^2 = 2(x^2+y^2)

Difference of two squares → factor it, never square the big numbers.
$(x+y)^2 + (x-y)^2 = 2x^2 + 2y^2$ ; their difference is $4xy$ .

Why: the cross terms ( $\pm 2xy$ ) cancel or double depending on the combination.

Example: Find $105^2 - 95^2$ .

$(105-95)(105+95) = 10 \times 200 = 2000$ .

Type 3: x + 1/x family (squares, cubes, fourth powers)very common2 practice Q

How to spot it:

'If $x + \frac{1}{x} = 4$ , find $x^2 + \frac{1}{x^2}$ ' — the minus version is equally common.

x^2+\tfrac{1}{x^2} = k^2 \mp 2,\quad x^3\pm\tfrac{1}{x^3} = k^3 \mp 3k

Square the given sum — the cross term is always 2.
Subtract 2 (plus version) or add 2 (minus version).
For cubes use $k^3 \mp 3k$ ; for fourth powers, square the squared result and subtract 2.

Why: $\left(x \pm \frac1x\right)^2 = x^2 + \frac{1}{x^2} \pm 2$ .

Example: If $x - \frac{1}{x} = 5$ , find $x^2 + \frac{1}{x^2}$ .

$25 + 2 = 27$ .

Type 4: Given a + b and ab: find a² + b², a³ + b³very common2 practice Q

How to spot it:

The pair sum and product are given; the target is a square-sum, cube-sum or similar symmetric value.

a^2+b^2 = (a+b)^2-2ab,\quad a^3+b^3 = (a+b)^3-3ab(a+b)

Write the target in terms of (a+b) and ab.
Substitute the two given numbers.
If asked for a and b themselves, spot small integer pairs.

Why: every symmetric expression in a and b is a function of (a+b) and ab.

Example: If $a+b = 7$ , $ab = 12$ , find $a^3+b^3$ .

$343 - 3 \times 12 \times 7 = 91$ (the numbers are 3 and 4).

Type 5: Zero-sum cubes: a³ + b³ + c³ = 3abccommon2 practice Q

How to spot it:

Three cubes (fractions or signed numbers) whose bases add up to 0.

a+b+c = 0 \Rightarrow a^3+b^3+c^3 = 3abc

Add the three bases — confirm the sum is 0.
Multiply the bases together, then by 3.
Mind the signs: two negatives multiply to a plus.

Why: $a^3+b^3+c^3 - 3abc = (a+b+c)(\dots)$ , so a zero sum forces the cube-sum to equal $3abc$ .

Example: Find $25^3 + (-17)^3 + (-8)^3$ .

Bases: $25-17-8 = 0$ → $3 \times 25 \times 17 \times 8 = 10200$ .

Formulas

Square of sum / difference

(a \pm b)^2 = a^2 \pm 2ab + b^2

Difference of squares

a^2 - b^2 = (a+b)(a-b)

Sum of cubes

a^3 + b^3 = (a+b)(a^2 - ab + b^2)

Difference of cubes

a^3 - b^3 = (a-b)(a^2 + ab + b^2)

Cube of sum

(a+b)^3 = a^3 + b^3 + 3ab(a+b)

Three cubes

a^3 + b^3 + c^3 - 3abc = (a+b+c)(a^2+b^2+c^2-ab-bc-ca)

Zero-sum case

a + b + c = 0 \Rightarrow a^3 + b^3 + c^3 = 3abc

Shortcut tricks

⚡ Pattern-match the fraction

If the numerator has cubes and the denominator has three terms with a middle product, it is a sum/difference-of-cubes identity.

Example: Simplify $\frac{0.8 \times 0.8 \times 0.8 - 0.5 \times 0.5 \times 0.5}{0.8 \times 0.8 + 0.8 \times 0.5 + 0.5 \times 0.5}$ .

Form $\frac{a^3 - b^3}{a^2 + ab + b^2} = a - b$ with $a = 0.8$ , $b = 0.5$ ⇒ $0.3$ .

⚡ a + b + c = 0 ⇒ cubes = 3abc

Check if the three bases add to zero before cubing anything.

Example: Find $(-12)^3 + 7^3 + 5^3$ .

$-12 + 7 + 5 = 0$ ⇒ value $= 3 \times (-12) \times 7 \times 5 = -1260$ .

⚡ Products around a round number

$(n - d)(n + d) = n^2 - d^2$ .

Example: Find $998 \times 1002$ .

$1000^2 - 2^2 = 1000000 - 4 = 999996$ .

Where students lose marks

Using $a^2 + ab + b^2$ with the sum-of-cubes numerator (sign of the middle term must be opposite).
Expanding decimals by brute force instead of naming a and b.
Writing $(a+b)^3 = a^3 + b^3$ .

Practice sets — 12 questions

Sets of 10, mixed across the question types above. Each answer comes with a step-by-step explanation.

Topic test · 12 questions

Suggested time 8 min · wrong answers go to your mistake notebook automatically.