SSC CGL Maths · Surds, Indices and Simplification

Simplification Shortcuts

Many long-looking expressions are an identity or a fixed ratio in disguise; seeing the pattern replaces the arithmetic, and a digit-sum or last-digit check confirms the option in seconds.

All Surds, Indices and Simplification notes

Why this matters

Eighteen questions in the bank, 11 EASY and 7 MODERATE, and 13 of them come from the 2025 shifts. Ten are one pattern: a sum of cubes divided by a sum of cubes whose terms are a fixed multiple of the first. Spot the multiple and the answer is one cube, with no cubing of decimals at all.

Concept 1 of 3: Difference of squares and cubes as a shortcut

An expression like

7.2 \times 7.2 - 2.8 \times 2.8

is the identity

a^2 - b^2

with numbers in place of letters. Recognise the shape and the factors do the work: one easy product instead of two hard squares.

Definition

$a^2 - b^2 = (a + b)(a - b)$ , so $\frac{a^2 - b^2}{a + b} = a - b$ .
$a^3 - b^3 = (a - b)(a^2 + ab + b^2)$ and $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$ .
A product written out as $a \times a \times a$ is a cube in disguise; write it as $a^3$ to see the pattern.
Square a decimal by splitting it: $(a + b)^2 = a^2 + 2ab + b^2$ , so $20.3^2 = 400 + 12 + 0.09 = 412.09$ .
These identities are taught in full in the Algebraic Identities chapter; here they are used only to cut arithmetic.

Square and cube identities

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

Worked example

Simplify

\frac{8.6 \times 8.6 - 3.4 \times 3.4}{0.52}

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2023 · Tier I, 27 Jul 2023, Shift 3 · Q67Easy

Example 1 · Surds, Indices and Simplification · Simplification Shortcuts

Simplify the following expression.

\frac{7.35 \times 7.35 - 2.25 \times 2.25}{0.24}

Pairing a cube with the wrong middle sign

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

, but

\frac{a^3 - b^3}{a^2 - ab + b^2}

does not simplify. The middle term is plus for

a^3 - b^3

and minus for

a^3 + b^3

; check it before using the shortcut.

Dropping the middle term of a square

(30 + 0.5)^2

900 + 30 + 0.25 = 930.25

, not

900.25

. The term

2ab

is the one people forget.

Drill 3 more on difference of squares and cubes as a shortcut

Concept 2 of 3: Sums of cubes in a fixed ratio

If every term below is the same multiple

k

of the matching term above, then every cube below is

k^3

times the cube above. So the whole bottom is

k^3

times the whole top, and nothing needs cubing.

Definition

If $c = ka$ and $d = kb$ , then $\frac{a^3 + b^3}{c^3 + d^3} = \frac{a^3 + b^3}{k^3(a^3 + b^3)} = \frac{1}{k^3}$ .
It works for any number of terms, and for squares too (the value is then $\frac{1}{k^2}$ ), as long as one k links every pair.
Find k by dividing one pair, then check that the other pair gives the same k.
If the top terms are the bigger ones, the value is $k^3$ rather than $\frac{1}{k^3}$ .
If one pair gives a different k, the shortcut does not apply: work the cubes out.
Cubes to know: $2^3 = 8$ , $3^3 = 27$ , $4^3 = 64$ , $5^3 = 125$ , and $0.5^3 = 0.125$ , $0.2^3 = 0.008$ , $0.1^3 = 0.001$ .

Sums of cubes in a fixed ratio

\frac{a^3 + b^3}{(ka)^3 + (kb)^3} = \frac{1}{k^3}

kthe multiple linking each term below to its partner above

Worked example

Find

\frac{0.3^3 + 0.08^3}{1.2^3 + 0.32^3}

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 18 Sep 2025, Shift 1 · Q74Easy

Example 2 · Surds, Indices and Simplification · Simplification Shortcuts

What is the value of

(0.12^3 + 0.06^3) \div (0.24^3 + 0.12^3)

Using k instead of k cubed

If each term below is 5 times its partner above, the bottom is

5^3 = 125

times the top, so the value is

\frac{1}{125} = 0.008

, not

\frac{1}{5} = 0.2

Pairing terms by position

Match each term with the one it is a multiple of, not the one in the same place. In

\frac{0.6^3 + 0.2^3}{0.1^3 + 0.3^3}

the 0.6 pairs with 0.3 and the 0.2 with 0.1:

k = 2

both times, so the value is 8.

Drill 9 more on sums of cubes in a fixed ratio

Concept 3 of 3: Common factors and quick answer checks

Before multiplying big numbers, look for a factor they share. Before trusting a long calculation, test the options with a check that takes seconds: the last digit, the remainder on division by 9, the sign.

Definition

Take out a common factor: $ab + ac = a(b + c)$ , so $17 \times 46 + 17 \times 54 = 17 \times 100 = 1700$ .
Cancel before you multiply: break each number into prime factors and strike out the factors shared by top and bottom.
Repeated-digit sums: $3 + 33 + 333 = 3 \times (1 + 11 + 111) = 3 \times 123 = 369$ .
Digit-sum check: a number leaves the same remainder on division by 9 as the sum of its digits. Find the remainder of each part, combine with the same signs, and keep the option whose digit sum matches.
Last-digit check: the last digit of a sum or product depends only on the last digits.
These checks rule options out; they do not prove the one left. If two options pass, work the sum.

Common factor

ab + ac = a(b + c) \qquad ab - ac = a(b - c)

Worked example

Which of 7860, 8280, 8000 and 8380 is the value of

23^3 - 15^3 - 8^3

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 12 Sep 2025, Shift 1 · Q75Moderate

Example 3 · Surds, Indices and Simplification · Simplification Shortcuts

31^3 + 18^3 - 37^3 + 210

is equal to:

Treating a passed check as proof

Two options can share a last digit or a digit sum. A check removes the options that fail it; it confirms the one left only when it is the only one.

Losing the sign in a remainder check

A subtraction stays a subtraction: if the parts leave 1 and 4, their difference leaves

1 - 4 = -3

, which is the same as a remainder of 6 on division by 9.

Drill 3 more on common factors and quick answer checks

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (3)

Difference of squares and cubes as a shortcut
Square and cube identities
$a^2 - b^2 = (a + b)(a - b) \qquad a^3 - b^3 = (a - b)(a^2 + ab + b^2) \qquad a^3 + b^3 = (a + b)(a^2 - ab + b^2) \qquad (a + b)^2 = a^2 + 2ab + b^2$
Sums of cubes in a fixed ratio
Sums of cubes in a fixed ratio
$\frac{a^3 + b^3}{(ka)^3 + (kb)^3} = \frac{1}{k^3}$
Common factors and quick answer checks
Common factor
$ab + ac = a(b + c) \qquad ab - ac = a(b - c)$

Watch out for (6)

Pairing a cube with the wrong middle sign
→ Difference of squares and cubes as a shortcut
Dropping the middle term of a square
→ Difference of squares and cubes as a shortcut
Using k instead of k cubed
→ Sums of cubes in a fixed ratio
Pairing terms by position
→ Sums of cubes in a fixed ratio
Treating a passed check as proof
→ Common factors and quick answer checks
Losing the sign in a remainder check
→ Common factors and quick answer checks

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