Square = number times itself
Squaring \(a\) means multiplying it by itself.
\[a^2=a\times a\]
Example: \(7^2=49\).
2 SSC CGL · Mathematics
Square Roots & Elementary Surds
Handwritten-style notes for SSC CGL Tier I and Tier II.
01MAP IT!
The Big Picture — Graphic Mind Map
Square-root and surd questions move from recognition to simplification, then comparison or rationalisation.
ROOTS
&
SURDS
&
SURDS
1. Perfect Squaresrecognition, unit digits and ranges
2. Finding Rootsfactorisation, long division and decimals
3. Surd Languageorder, pure, mixed, like and unlike
4. Operationssimplify, add, multiply and divide
5. Rationalisationmonomial and conjugate denominators
6. Exam Speedcomparison, approximation and traps
\[\text{recognise}\ \longrightarrow\ \text{factor}\ \longrightarrow\ \text{simplify}\ \longrightarrow\ \text{check the domain and sign}\]
02FOUNDATION
The Foundation (Basics) — Core Concepts
Begin with the meaning of a square, then build every surd rule from multiplication.
ASquare and square root — the first idea
Square root reverses squaring
The principal square root of \(x\ge0\) is the non-negative number whose square is \(x\).
\[\sqrt{x}=a\iff a^2=x,\ a\ge0\]
Do not confuse these
\[\sqrt{49}=7\quad\text{but}\quad x^2=49\Rightarrow x=\pm7\]
The radical symbol denotes the principal root; an equation may have two roots.
Why is it called a “square”?
A square of side \(a\) contains area \(a^2\). Reversing the question—“which side gives this area?”—produces \(\sqrt{a^2}=|a|\).
\[\sqrt{a^2}=|a|\]
BPerfect squares and their fingerprints
Prime-exponent test
If \(N=\prod p_i^{\alpha_i}\), then
\[N\text{ is a perfect square}\iff 2\mid\alpha_i\ \text{for every }i\]
Possible last digits
A decimal square can end only in
\[0,1,4,5,6,9\]
It can never end in \(2,3,7,8\).
Zero and parity tests
- Trailing zeros occur in an even count.
- Square of an even number is divisible by \(4\).
- Square of an odd number is odd and congruent to \(1\pmod8\).
| Pattern | Reliable conclusion | Converse warning |
|---|---|---|
| Unit digit \(2,3,7,8\) | Definitely not a perfect square. | Ending in \(0,1,4,5,6,9\) does not guarantee a square. |
| Odd number of trailing zeros | Not a non-zero perfect square. | Even trailing zeros alone are insufficient. |
| Digital root | A square has digital root \(1,4,7\) or \(9\). | These digital roots are filters, not proofs. |
| Between consecutive squares | Exactly \(2n\) integers lie strictly between \(n^2\) and \((n+1)^2\). | The numerical gap is \(2n+1\). |
CFinding square roots by prime factorisation
1. FactorWrite the number as primes
2. PairMake equal prime pairs
3. PickTake one from each pair
4. MultiplyMultiply picked primes
5. VerifySquare the answer
Exact root
\[1764=2^2\times3^2\times7^2\]
\[\sqrt{1764}=2\times3\times7=42\]
Least multiplier or divisor
Unpaired prime powers reveal the repair.
\[540=2^2\times3^3\times5\]
Multiply by \(3\times5=15\), or divide by \(3\times5=15\), to make all exponents even.
Pair the prime “socks”
A complete pair comes out of the radical as one factor. Any unpaired prime stays inside.
\[\sqrt{p^{2k}}=p^k\quad(p\ge0)\]
DLong-division method — roots without factorisation
Grouping rule
From the decimal point, pair digits outward in both directions. Each digit-pair produces one root digit.
\[12\,|\,96\,|\,64\quad\Longrightarrow\quad\sqrt{129664}=360\]
Trial-digit rule
After bringing down the next pair, double the current root \(R\). Choose the largest digit \(x\) satisfying
\[(20R+x)x\le\text{current dividend}\]
The long-division “root machine”
- Pair digits.
- Choose the largest initial square.
- Subtract and bring down a pair.
- Double the root, attach trial digit, multiply.
- Repeat; add pairs of zeros for decimals.
EEstimating roots and decimal roots
Locate between squares
\[n^2\lt N\lt(n+1)^2\Rightarrow n\lt\sqrt N\lt n+1\]
Because \(49\lt60\lt64\), \(7\lt\sqrt{60}\lt8\).
Nearest-square approximation
For small \(h\) relative to \(a^2\):
\[\sqrt{a^2+h}\approx a+\frac{h}{2a}\]
This is an approximation, not an identity.
Decimals and fractions
\[\sqrt{0.0081}=0.09\]
\[\sqrt{\frac ab}=\frac{\sqrt a}{\sqrt b}\quad(a\ge0,b>0)\]
FWhat is a surd?
Definition
A surd is an irrational root written exactly, such as \(\sqrt2\), \(\sqrt[3]{5}\) or \(2+\sqrt3\).
Not every radical is a surd: \(\sqrt9=3\).
General form and order
\[\sqrt[n]{a}=a^{1/n}\]
The index \(n\) is the order. For even \(n\), real-valued radicand requires \(a\ge0\).
Pure and mixed surds
Pure: \(\sqrt{12}\). Mixed: \(2\sqrt3\).
\[a\sqrt b=\sqrt{a^2b}\quad(a\ge0)\]
Radical is notation; surd is a value-type
The radical symbol is only the root sign. After simplification, the value may be rational or irrational. SSC often tests this distinction.
GSimplification and standard form
Extract perfect powers
\[\sqrt{ab}=\sqrt a\sqrt b\quad(a,b\ge0)\]
\[\sqrt{72}=\sqrt{36\cdot2}=6\sqrt2\]
For higher roots
\[\sqrt[n]{a^nb}=a\sqrt[n]b\]
For even \(n\), use \(|a|\) in a fully real-domain statement.
Do not split addition
\[\sqrt{a+b}\ne\sqrt a+\sqrt b\]
Example: \(\sqrt{9+16}=5\ne7\).
HOperations with like and unlike surds
Add/subtract like surds
\[a\sqrt m\pm b\sqrt m=(a\pm b)\sqrt m\]
First simplify: \(\sqrt8+\sqrt{18}=2\sqrt2+3\sqrt2=5\sqrt2\).
Multiply
\[(a\sqrt m)(b\sqrt n)=ab\sqrt{mn}\]
\[(\sqrt a+\sqrt b)^2=a+b+2\sqrt{ab}\]
Conjugate identities
\[(a+b)(a-b)=a^2-b^2\]
\[(\sqrt a+\sqrt b)(\sqrt a-\sqrt b)=a-b\]
Like surds are like identical classroom labels
Coefficients may combine only when the simplified radical parts match. Simplify before deciding whether terms are like.
\[3\sqrt5+2\sqrt5=5\sqrt5\quad\text{but}\quad\sqrt3+\sqrt5\text{ stays as written}\]
IRationalisation — remove surds from the denominator
Single surd denominator
\[\frac{a}{\sqrt b}=\frac{a\sqrt b}{b}\quad(b>0)\]
Multiply numerator and denominator by the same surd.
Binomial denominator
Multiply by the conjugate.
\[\frac1{a+\sqrt b}=\frac{a-\sqrt b}{a^2-b}\]
Two-surd denominator
\[\frac1{\sqrt a+\sqrt b}=\frac{\sqrt a-\sqrt b}{a-b}\]
Valid when \(a\ne b\) and real roots exist.
The conjugate is a mirror partner
It changes only the middle sign. Multiplying the pair creates a difference of squares, so the radical cross-terms cancel.
\[(a+\sqrt b)(a-\sqrt b)=a^2-b\]
JComparison, ordering and equations
Positive roots preserve order
\[a,b\ge0:\quad \sqrt a\lt\sqrt b\iff a\lt b\]
Compare coefficients
For positive quantities, square both sides.
\[a\sqrt m\ ?\ b\sqrt n\Longleftrightarrow a^2m\ ?\ b^2n\]
Nested-radical identity
\[\sqrt{a\pm2\sqrt b}=\sqrt m\pm\sqrt n\]
where \(m+n=a\) and \(mn=b\).
Equation safety
Squaring can introduce extraneous roots. Isolate the radical, square, solve, then substitute into the original equation.
\[\sqrt{x+1}=x-1\Rightarrow x\ge1\]
Domain rules
- Even root: radicand must be non-negative in real numbers.
- Denominator must never be zero.
- Odd roots accept negative radicands.
\[\sqrt[3]{-8}=-2\]
KFormula and value bank
| Bank | Values / identities |
|---|---|
| Squares \(1^2\)–\(15^2\) | \(1,4,9,16,25,36,49,64,81,100,121,144,169,196,225\) |
| Squares \(16^2\)–\(30^2\) | \(256,289,324,361,400,441,484,529,576,625,676,729,784,841,900\) |
| Approximate roots | \(\sqrt2\approx1.414,\ \sqrt3\approx1.732,\ \sqrt5\approx2.236,\ \sqrt7\approx2.646,\ \sqrt{10}\approx3.162\) |
| Core identities | \((\sqrt a+\sqrt b)^2=a+b+2\sqrt{ab}\); \((\sqrt a-\sqrt b)^2=a+b-2\sqrt{ab}\) |
| Reciprocal conjugates | \((\sqrt a+\sqrt b)^{-1}=(\sqrt a-\sqrt b)/(a-b)\), for \(a\ne b\) |
03SAVE TIME
Short Tricks & Magic Formulas
Use the shortcut only after checking its sign, domain and “perfect-square” condition.
1Near-square mental root
Exact difference
\[(n+1)^2-n^2=2n+1\]
Move between adjacent squares without full multiplication.
Numbers ending in \(5\)
\[(10a+5)^2=100a(a+1)+25\]
For \(65^2\): \(6\times7=42\), append \(25\), giving \(4225\).
Square around a base
\[(a\pm b)^2=a^2\pm2ab+b^2\]
Example: \(98^2=(100-2)^2=9604\).
2Fast root-digit and range checks
Number of root digits
If a perfect square has \(d\) digits, its positive square root has
\[\left\lceil\frac d2\right\rceil\text{ digits}\]
Unit-digit pairs
\[1\leftrightarrow1,9;\quad4\leftrightarrow2,8;\quad5\leftrightarrow5;\quad6\leftrightarrow4,6\]
Use the leading digits to choose between paired candidates.
Count squares in an interval
\[\#=\lfloor\sqrt R\rfloor-\lceil\sqrt L\rceil+1\]
Counts perfect squares in \([L,R]\).
3Comparison without decimals
Square positive expressions
\[3\sqrt5\ ?\ 2\sqrt{11}\Longrightarrow45\ ?\ 44\]
Therefore \(3\sqrt5>2\sqrt{11}\).
Shift a binomial
To compare \(\sqrt a+\sqrt b\), isolate one radical before squaring; do not square blindly when signs are unknown.
Reciprocal pair
\[(\sqrt a+\sqrt b)(\sqrt a-\sqrt b)=a-b\]
If \(a-b=1\), the conjugate expressions are reciprocals.
4Surd-expression power tricks
When product is \(1\)
If \(x=\sqrt a+\sqrt{a-1}\), then
\[\frac1x=\sqrt a-\sqrt{a-1}\]
\[x+\frac1x=2\sqrt a\]
Nested radical recognition
For \(\sqrt{a+2\sqrt b}\), find two numbers with sum \(a\) and product \(b\).
\[\sqrt{7+4\sqrt3}=2+\sqrt3\]
SSC speed dashboard
Before calculating, test four filters: possible unit digit, neighbouring squares, sign/domain, and whether a conjugate will cancel the radical.
04DON'T RUSH
The SSC / TCS Traps — Red Flags 🚩
Most wrong options are built from one illegal radical step or a missed sign condition.
Trap 1: principal root vs equation roots
- \(\sqrt{49}=7\), never \(\pm7\).
- But solving \(x^2=49\) gives \(x=\pm7\).
- Also \(\sqrt{x^2}=|x|\), not always \(x\).
Trap 2: illegal distribution
\[\sqrt{a+b}\ne\sqrt a+\sqrt b\]
The product rule needs non-negative real factors; the sum rule does not exist.
Trap 3: simplify before judging
- Apparently unlike surds may become like after extraction.
- Rationalise the complete denominator using its conjugate.
- After squaring an equation, verify every solution in the original.
05LOCK IT IN
Memory Hooks & Mnemonics
Attach the algebraic rule to a picture that can be recalled under exam pressure.
“Squares wear pairs”
A perfect square has even prime exponents. One prime from every pair emerges from the root.
“Perfect pairs exit the radical gate”
Extract complete square factors; unfinished pairs remain inside.
“Conjugates change one sign, not the design”
The mirror pair keeps every term and flips only the middle sign.
“Root sign chooses the positive door”
The principal square root is non-negative; equations may reopen the negative door.
“Like labels can combine”
Only simplified surds with identical radical parts add or subtract through coefficients.
“Range first, digits second”
Neighbouring squares fix the size; the unit digit then selects the candidate.
Principal rootNon-negative
Perfect squareEven prime exponents
Like surdsSame simplified radical part
ConjugateFlip the middle sign
RationaliseRemove radicals below
Final checkDomain, sign and substitution