Squares, Square Roots, Indices & Surds Notes

Last reviewed 16 Sept 2026 · 6 min read

Squares and cubes

Unit digit of a perfect square can only be 0, 1, 4, 5, 6 or 9 (never 2, 3, 7, 8).
A perfect square ending in 0 has an even number of zeros.
Squares of even numbers are even; of odd numbers are odd.
Number of factors of a perfect square is odd.
Digit sum (mod 9) of a perfect square is 1, 4, 7 or 9.

FormulaSquaring shortcuts

Ending in 5: $(10 a + 5)^{2} = a (a + 1) \times 100 + 25$ → $6 5^{2} = 4225$
Near 50: $(50 \pm x)^{2} = (25 \pm x) \times 100 + x^{2}$ → $4 7^{2} = 22 \times 100 + 9 = 2209$
Near 100: $(100 \pm x)^{2} = (100 \pm 2 x) \times 100 + x^{2}$ → $9 7^{2} = 94 \times 100 + 9 = 9409$ ; $10 4^{2} = 10816$
Identity: $a^{2} = (a + b) (a - b) + b^{2}$ → $4 8^{2} = 50 \times 46 + 4 = 2304$

Prime factorisation — pair equal factors; take one from each pair.
Long division method — pair digits from the decimal point; successive trial divisors.
Estimation — locate between known squares (e.g. √150 lies between 12 and 13, closer to 12: 12.25).

Prime factorisation — group factors in threes.
Perfect cubes (up to 6 digits) mental method: unit digit of cube determines unit digit of root (1→1, 8→2, 7→3, 4→4, 5→5, 6→6, 3→7, 2→8, 9→9, 0→0); remaining leading part (after removing last three digits) gives the tens digit.
- Example: ∛9261 → unit digit 1 → root ends in 1; leading part 9 lies between 8 (2³) and 27 (3³) → tens digit 2 → 21.

FormulaLaws of exponents

A surd is an irrational root of a rational number, e.g. $2$ , $35$ ( $4 = 2$ is not a surd).

Term	Meaning
Order	Index of the root ( $n a$ has order n)
Pure surd	Entire number under root — $12$
Mixed surd	Rational × surd — $23$
Like surds	Same irrational part — $32$ , $52$
Unlike surds	Different irrational parts — $2$ , $3$
Conjugate	$a + b$ and $a - b$

Add/subtract only like surds: $32 + 52 = 82$ .
Multiply: $a \times b = ab$ ; simplify $12 = 23$ .
Rationalising the denominator: multiply numerator and denominator by the conjugate:

\frac{1}{a + b} = \frac{a - b}{a - b}

Same order: compare the numbers under the roots.
Different orders: convert to a common order (LCM of indices) and compare radicands. E.g. √3 vs ∛5 → raise to power 6: $3^{3} = 27$ vs $5^{2} = 25$ → √3 > ∛5.

a + 2 b = x + y where x + y = a, x y = b

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