Number System & Simplification

Number system questions are the silent backbone of every RPF Sub-Inspector mathematics paper — primes, co-primes, even/odd, rationals — they all turn into one-line answers if you have the family tree clearly drawn in your head.

Definition: Natural numbers (N) are the counting numbers: 1, 2, 3, 4, … . They start from 1 and never include 0 in the Indian school convention used by RPF/SSC/Railway exams.

Definition: Whole numbers (W) are the natural numbers together with 0: 0, 1, 2, 3, … . They are exactly N with zero added.

Definition: Integers (Z) include all whole numbers and their negatives: …, −3, −2, −1, 0, 1, 2, 3, … . Z comes from the German Zahl meaning "number".

Definition: Rational numbers (Q) are numbers that can be written as p/q where p and q are integers and q ≠ 0. Every integer is a rational number (e.g., 5 = 5/1). Their decimal expansions either terminate (e.g., 0.75) or recur (e.g., 0.333…).

Definition: Irrational numbers cannot be written as p/q. Their decimal expansions are non-terminating and non-recurring. Examples: √2 ≈ 1.41421356…, π ≈ 3.14159265…, e ≈ 2.71828182….

Definition: Real numbers (R) are the union of all rationals and irrationals; every point on the number line is a real number.

Definition: A prime number is a natural number greater than 1 that has exactly two distinct positive factors — 1 and itself. The smallest prime is 2, and it is the only even prime.

Definition: A composite number is a natural number with more than two distinct positive factors (e.g., 4, 6, 8, 9, 10).

Definition: Co-prime (relatively prime) numbers are two numbers whose HCF (greatest common factor) is 1. They need not themselves be prime — for example, 8 and 15 are co-prime even though both are composite.

The number family tree, drawn once

Think of the number families as a series of larger and larger circles. The innermost circle is N (1, 2, 3…). Wrap a slightly bigger circle around it and add 0 — now you have W. Wrap another circle and add the negatives — that gives Z. Wrap one more and add all p/q forms — that gives Q. Outside Q live the irrationals. Q together with the irrationals is the universe R.

This nesting is testable directly: "Is −3 a whole number?" No, because W stops at 0 and goes upward; −3 lives in Z but not W. "Is 7 a rational number?" Yes, because 7 = 7/1. "Is √4 irrational?" No, because √4 = 2, which is a perfectly good integer.

Primes, composites and the one in the middle

Prime numbers are the building blocks of every natural number (Fundamental Theorem of Arithmetic): every integer greater than 1 factors uniquely into primes. The first few primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. There are exactly 25 primes below 100, a constant that RPF, SSC and Railway exams have asked verbatim more than once. Memorise the number, and ideally memorise the primes themselves up to 50.

The number 1 is the odd one out: it is neither prime nor composite. It has only one factor (itself), so it fails the "exactly two factors" rule for prime and the "more than two factors" rule for composite.

The number 2 is also special — it is the only even prime. Every other even number is divisible by 2 (apart from itself and 1) and is therefore composite.

Co-prime, not "both prime"

Co-prime asks only about the HCF. The pair (8, 15) is co-prime because HCF(8, 15) = 1, even though 8 and 15 are both composite. Conversely, two primes are always co-prime, but the converse is not required.

Why it matters: RPF and SSC question-setters love trap options like "co-prime means both numbers are prime". A single second of confusion here can cost you a mark.

Real-world example: A railway ticket counter shows ticket numbers that increment by either 7 or 9 on adjacent counters. Since HCF(7, 9) = 1, the two counters' ticket numbers are co-prime — and they will only repeat together after 7 × 9 = 63 tickets. The LCM = HCF × … identity behind this is the same one that makes co-prime so practically useful.

Common misconception: "Every even number is composite." Wrong — 2 is even and prime. Also wrong: "Every prime is odd." Wrong for the same reason.

Common misconception: "Recurring decimals like 0.333… are irrational." Wrong — they are rational, equal to 1/3.

Question: Classify each of the following: −5, 0, 1, 2, 4, √7, 22/7, 0.272727…

Solution:
Step 1: −5 — integer, rational, real. Not natural, not whole, not prime, not composite.
Step 2: 0 — whole, integer, rational, real. Not natural, not prime, not composite.
Step 3: 1 — natural, whole, integer, rational, real. Neither prime nor composite.
Step 4: 2 — natural, whole, integer, rational, real, and the only even prime.
Step 5: 4 — natural, whole, integer, rational, real, composite (factors 1, 2, 4).
Step 6: √7 — real, irrational (cannot be written as p/q).
Step 7: 22/7 — rational (it is a fraction). Note that 22/7 is not π; it is only an approximation.
Step 8: 0.272727… — rational, since the bar makes it 27/99 = 3/11.
Conclusion: Tag every number with as many family labels as fit; the smallest set is most specific.

:::compare

Set	Includes	Excludes	Smallest member
Natural (N)	1, 2, 3, …	0, negatives, fractions	1
Whole (W)	0, 1, 2, …	negatives	0
Integers (Z)	…, −2, −1, 0, 1, 2, …	fractions	no smallest
Rational (Q)	p/q form, terminating/recurring decimals	irrationals	no smallest
Irrational	√2, π, e	rationals	no smallest
Real (R)	rational + irrational	imaginary numbers	no smallest
:::

:::keypoints

• N ⊂ W ⊂ Z ⊂ Q ⊂ R (each set sits inside the next).
• 0 is whole but not natural.
• 1 is neither prime nor composite.
• 2 is the only even prime.
• Co-prime ⇔ HCF = 1 (they need not be prime).
• Even numbers end in 0, 2, 4, 6, 8; odd in 1, 3, 5, 7, 9.
• 25 prime numbers exist below 100.
• Recurring decimals are rational; non-recurring non-terminating decimals are irrational.
  :::

:::memory
"NW-Z-Q-R" with mnemonic *"Naughty Wily Zebras Quietly Rest"* gives the order of the families from smallest to largest. And "One-Twee": 1 is neither prime nor composite, 2 is the only even prime.
:::

:::recap

• The family tree grows N → W → Z → Q → R, each set strictly bigger than the last.
• Prime = exactly two factors; composite = more than two; 1 is in neither club.
• Co-prime means HCF = 1, nothing more.
• 25 primes lie below 100 — a fact the question paper expects you to know.
  :::

By 2: last digit even. By 3: digit-sum divisible by 3. By 4: last two digits divisible by 4. By 5: ends in 0 or 5. By 6: divisible by both 2 and 3. By 8: last three digits divisible by 8. By 9: digit-sum divisible by 9. By 10: ends in 0. By 11: (sum of odd-place digits) - (sum of even-place digits) = 0 or multiple of 11. By 25: last two digits 00,25,50,75. Shortcut for 7: double the last digit, subtract from the rest; if result divisible by 7, the number is too (e.g., 203: 20 - 6 = 14, divisible by 7).

Check if 4,832,718 is divisible by 11. Mark places from the right: digits 8(1),1(2),7(3),2(4),3(5),8(6),4(7). Odd-place (1,3,5,7) digits: 8+7+3+4 = 22. Even-place (2,4,6) digits: 1+2+8 = 11. Difference = 22 - 11 = 11, which is a multiple of 11, so the number IS divisible by 11. This alternating-sum method is far faster than long division and is the most-tested divisibility rule in RPF SI papers, so practice tagging odd/even positions from the rightmost digit.

HCF (Highest Common Factor) = the largest number dividing all given numbers. LCM (Lowest Common Multiple) = smallest number divisible by all. KEY RELATION: for any two numbers a and b, HCF x LCM = a x b. This lets you find one when three values are known. HCF is always less than or equal to the smallest number; LCM is always greater than or equal to the largest number. HCF always divides LCM exactly. For fractions: HCF of fractions = HCF(numerators)/LCM(denominators); LCM of fractions = LCM(numerators)/HCF(denominators). Memory aid: 'Highest Cuts Fractions' (HCF uses HCF on top), opposite for LCM.

Prime factorisation method: write each number as a product of primes. HCF = product of the LOWEST powers of common primes. LCM = product of the HIGHEST powers of all primes appearing. Example: 12 = 2^2 x 3, 18 = 2 x 3^2. HCF = 2^1 x 3^1 = 6; LCM = 2^2 x 3^2 = 36. Check: 6 x 36 = 216 = 12 x 18. The division (ladder) method is faster for LCM of many numbers: divide repeatedly by common primes, then multiply all divisors and remaining quotients. For HCF of large numbers, the long-division (Euclidean) method is quickest.

BELLS RINGING TOGETHER: if bells toll at intervals of 6, 9, 12 seconds, they ring together after LCM(6,9,12) = 36 seconds; in 1 hour (3600 s) together = 3600/36 + 1 times (the +1 counts the start). COMMON REMAINDER: 'smallest number which when divided by a, b, c leaves the same remainder r' = LCM(a,b,c) + r. 'Largest number dividing x, y, z leaving remainder r each' = HCF(x-r, y-r, z-r), or HCF of the differences when remainders are equal but unknown. These two templates cover the majority of RPF SI HCF/LCM word problems — identify which one the question fits first.

BODMAS fixes the sequence for any simplification: B - Brackets (solve innermost first: (), {}, []), O - Of / Order (powers, roots, 'of' means multiply), D - Division, M - Multiplication, A - Addition, S - Subtraction. Division and multiplication rank equally — work left to right; same for addition and subtraction. The word 'of' is treated as multiplication but is performed before normal division (e.g., 1/2 of 8 / 4 = 4/4 = 1). Memory aid: 'Brackets Order DM AS.' A single misstep in operation order is the most common reason candidates lose marks in the simplification section — always rewrite the expression bracket-by-bracket.

Some boards use VBODMAS where V = Vinculum (bar/line), solved first — e.g., in 12 - (8 - bar over 3-1), simplify the bar 3-1=2 first. For mixed fractions, convert to improper fractions before applying BODMAS. Cancel common factors early to avoid large numbers. For chained 'of' and division: 'of' binds tighter, so 24 / 4 of 3 = 24 / 12 = 2 (do the 'of' multiplication first). When you see surds or squares, evaluate Order before Division. Shortcut: replace repeated decimals like 0.333... with fractions (1/3) to keep arithmetic exact rather than approximate.

Simplify: 36 - [18 - {14 - (15 - 6 / 3 x 2)}]. Step 1 innermost: 6/3 = 2, then 2 x 2 = 4, so (15 - 4) = 11. Step 2 next bracket: {14 - 11} = 3. Step 3: [18 - 3] = 15. Step 4: 36 - 15 = 21. Answer = 21. Note how division and multiplication inside the parentheses were done left to right BEFORE the subtraction. Working strictly from the innermost bracket outward prevents sign errors — a frequent trap in RPF SI multi-bracket simplification problems.

A surd-and-indices question in RPF SI takes about thirty seconds — if you remember the eight laws cold. Otherwise it eats two whole minutes and lures you into wrong answers that look right. So treat this lesson as muscle memory, not as theory: every law below must come to your fingertips without thought.

Definition: An index (plural: indices) or exponent is the small number written above a base that tells you how many times the base multiplies itself. In a^n, "a" is the base and "n" is the exponent.

Definition: A surd is an irrational root that cannot be simplified to a rational number — for example, sqrt(2) or cube-root(5). Surds are expressed in radical form, but for arithmetic they are usually converted to fractional powers.

The eight laws — your toolbox

These eight identities are the engine of nearly every RPF SI question on simplification, surds and indices. Internalise them so well that you can apply them backwards as well as forwards.

1. Product of powers, same base — a^m x a^n = a^(m+n). When the base is the same, you add the exponents.
2. Quotient of powers, same base — a^m / a^n = a^(m-n). When dividing, you subtract the exponents.
3. Power of a power — (a^m)^n = a^(mn). When raising a power to another power, you multiply the exponents.
4. Power of a product — (ab)^n = a^n x b^n. The exponent distributes across multiplication.
5. Power of a quotient — (a/b)^n = a^n / b^n. The exponent distributes across division.
6. Zero exponent — a^0 = 1, for any a not equal to 0. Anything to the power zero is one.
7. Negative exponent — a^(-n) = 1 / a^n. A negative sign on the exponent flips the base into a reciprocal.
8. Fractional exponent — a^(m/n) = n-th root of a^m. The denominator is the root and the numerator is the power.

Why each law works — quick intuition

Law 1 is just counting. a^3 x a^2 means (a x a x a) x (a x a) — which is a multiplied by itself five times, i.e. a^5. Five is three plus two. So same base, add exponents.

Law 2 is the reverse. a^5 / a^2 means (a x a x a x a x a) / (a x a). Two factors cancel, three remain, giving a^3. So same base divided, subtract exponents.

Law 3 follows from Law 1. (a^3)^2 means a^3 x a^3, which by Law 1 is a^(3+3) = a^6 = a^(3 x 2). So power of a power, multiply.

Laws 4 and 5 follow from the commutativity of multiplication. (ab)^3 = ab x ab x ab = (a x a x a) x (b x b x b) = a^3 x b^3.

Law 6 comes from Law 2 with m = n. a^n / a^n = a^(n-n) = a^0. But anything divided by itself is 1. So a^0 = 1.

Law 7 comes from extending Law 2 below zero. a^0 / a^n = a^(-n). But a^0 = 1 and a^0 / a^n = 1 / a^n. So a^(-n) = 1 / a^n.

Law 8 ties surds to indices. By Law 3, (a^(1/2))^2 = a^(2/2) = a^1 = a. So a^(1/2) is the number whose square is a — i.e. the square root of a. By the same argument, a^(1/n) is the n-th root.

The strategy for comparison questions

To compare two powers, the standard RPF SI trick is to make either the base or the exponent equal — then compare what's left.

If the bases can be brought to the same value, raise both to whatever power is needed; once bases match, just compare exponents.

If the exponents can be matched, do it the other way — once exponents match, just compare bases.

Question: Which is larger, 2^30 or 3^20?

Solution:

Step 1: Bases differ; can we match exponents? GCD of 30 and 20 is 10. So write each as something to the power 10.

Step 2: 2^30 = (2^3)^10 = 8^10. Similarly 3^20 = (3^2)^10 = 9^10.

Step 3: Both are to the power 10 — now compare bases. 9 > 8.

Conclusion: 9^10 > 8^10, so 3^20 > 2^30.

Roots as fractional powers

The single most powerful habit in surd questions: rewrite roots as fractional powers, then apply the eight laws.

Examples you should be able to do in your head:

• sqrt(a) = a^(1/2)
• cube-root(a) = a^(1/3)
• a x sqrt(a) = a^1 x a^(1/2) = a^(3/2)
• sqrt(a) / cube-root(a) = a^(1/2) / a^(1/3) = a^(1/2 - 1/3) = a^(1/6)

Question: Simplify sqrt(2) x cube-root(4) x sixth-root(32).

Solution:

Step 1: Rewrite as fractional powers of 2.

• sqrt(2) = 2^(1/2).
• cube-root(4) = (2^2)^(1/3) = 2^(2/3).
• sixth-root(32) = (2^5)^(1/6) = 2^(5/6).

Step 2: Combine using Law 1.

2^(1/2 + 2/3 + 5/6). LCM of 2, 3, 6 is 6. So 1/2 = 3/6, 2/3 = 4/6, 5/6 = 5/6. Sum = 12/6 = 2.

Step 3: Total = 2^2 = 4.

Conclusion: The simplified value is 4. A messy-looking surd reduces to a clean integer the moment you switch to fractional powers.

Why it matters

Why it matters: Roughly 60-70% of RPF SI surd-and-indices questions can be cleared in under a minute by spotting one of these laws and applying it once or twice. Without the laws, even simple questions force you into clumsy expansion — burning time and creating arithmetic errors.

Real-world example

Real-world example: when you read in a science article that "the brightness of a star falls off as 1 over distance squared," that is a^(-2) in everyday English — Law 7 at work. When a bank computes compound interest, the formula uses (1 + r)^n — a Law-3 expression. The laws of indices show up in physics, finance and computer science precisely because they are the basic language of repeated multiplication.

Common misconception

Common misconception: Many candidates write (a + b)^n = a^n + b^n. This is false. The distributive law applies only to multiplication and division (Laws 4 and 5), not to addition. Try a = 1, b = 1, n = 2: (1+1)^2 = 4, but 1^2 + 1^2 = 2. The correct expansion is the binomial theorem — but for RPF SI you only need to remember: never split a power over a sum.

Another slip-up: writing 0^0 = 1 in answer sheets. The law a^0 = 1 explicitly excludes a = 0. The expression 0^0 is indeterminate, and in exam settings it does not appear in valid questions.

The standard simplification template

Use this checklist on every surd-and-indices question:

• Convert all roots to fractional powers.
• Express the bases in their smallest prime factors (e.g., 8 → 2^3, 32 → 2^5).
• Apply Law 1 / Law 2 to combine same-base powers.
• Apply Law 3 to collapse stacked powers.
• Convert negative exponents to reciprocals if the answer choices use them.
• Simplify the final exponent fraction to lowest terms.

:::compare

Operation	Law	Example
Multiplication, same base	a^m x a^n = a^(m+n)	2^3 x 2^4 = 2^7
Division, same base	a^m / a^n = a^(m-n)	5^6 / 5^2 = 5^4
Power of a power	(a^m)^n = a^(mn)	(3^2)^3 = 3^6
Power of a product	(ab)^n = a^n x b^n	(2 x 5)^3 = 2^3 x 5^3
Power of a quotient	(a/b)^n = a^n / b^n	(3/4)^2 = 9/16
Zero exponent	a^0 = 1 (a not 0)	7^0 = 1
Negative exponent	a^(-n) = 1/a^n	2^(-3) = 1/8
Fractional exponent	a^(m/n) = (n-th root of a)^m	27^(2/3) = 9
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:::keypoints

• Eight laws govern every surd-and-indices question — memorise them cold.
• Same base, multiplication: ADD the exponents.
• Same base, division: SUBTRACT the exponents.
• Power of a power: MULTIPLY the exponents.
• a^0 = 1 for a not equal to zero; 0^0 is indeterminate.
• A negative exponent flips the base into a reciprocal.
• Roots become fractional powers — sqrt(a) = a^(1/2), cube-root(a) = a^(1/3).
• To compare powers, match either the base or the exponent.
  :::

:::memory
Use the chant: "Same base ADD/SUBTRACT, power-of-power MULTIPLY." And for surds, the one-line rule: "Roots are fractional powers — convert and combine." For sign tricks: "Negative exponent flips it; zero exponent kills it (to 1)."
:::

:::recap

• The eight laws are the toolbox; everything else is one or two applications of them.
• Convert surds to fractional powers immediately — most problems collapse.
• Compare powers by matching the base or matching the exponent.
• Never split a power over a sum — the law applies only to products and quotients.
  :::

Surds look intimidating at first sight — a stubborn √2 stuck in a denominator can feel like the question is daring you to use a calculator. In RPF Sub-Inspector Mathematics, however, surds are designed to be tamed by two clean operations: simplification to lowest form and rationalisation of the denominator. Master them and what looks like 90 seconds of work becomes a 15-second answer.

Definition: A surd is a root of a positive rational number that cannot itself be expressed as a rational number. Examples: √2, √3, ∛5, ∜7. By contrast, √4 = 2 is not a surd because the answer is rational.

Why surds matter in the RPF SI paper

The RPF SI quant section, like SSC CGL, includes a small but consistent set of simplification questions where the only obstacle is an irrational root in a denominator. The marks are easy if you spot the standard manoeuvre and follow it without panic. The same techniques also appear inside speed-time, mensuration (think √(area) of a square) and Pythagoras questions.

Basic surd rules

These four laws cover almost every surd manipulation you will meet:

1. Multiplication: √a × √b = √(ab)
2. Division: √a / √b = √(a/b)
3. Power: (√a)² = a
4. Combining like surds: m√a ± n√a = (m ± n)√a (only when the radicands are the same)

These laws look obvious but watch the conditions — they hold for non-negative reals under square roots, and the “combining” rule needs the same number under the root. You cannot add √2 + √3 to get √5, because the radicands differ.

Simplifying a surd to lowest form

Definition: A surd is in lowest form when the number under the root has no perfect-square factor greater than 1.

The technique is to break the radicand into its largest perfect-square factor times the rest, then pull the square root of the perfect-square factor outside.

Example: √50 = √(25 × 2) = √25 × √2 = 5√2.

Other quick mental conversions to memorise:

• √8 = 2√2
• √12 = 2√3
• √18 = 3√2
• √27 = 3√3
• √32 = 4√2
• √45 = 3√5
• √48 = 4√3
• √72 = 6√2
• √75 = 5√3
• √98 = 7√2

For cube roots: ∛54 = ∛(27 × 2) = 3∛2. Look for perfect-cube factors (8, 27, 64, 125…) instead of perfect squares.

Rationalising the denominator

Definition: Rationalisation is the process of removing surds from the denominator of a fraction by multiplying numerator and denominator by a chosen factor that converts the denominator into a rational number.

The choice of multiplier depends on the form of the denominator. The two standard cases:

Case 1 — single surd in the denominator. For a denominator of the form √a, multiply by √a / √a.

Example: 1/√7 × √7/√7 = √7/7.

Case 2 — binomial surd in the denominator (a ± √b or √a ± √b). Multiply by the conjugate: change the sign between the two terms so that the product becomes a difference of squares.

The conjugate of (a + √b) is (a − √b), giving (a + √b)(a − √b) = a² − b — a clean rational number.

Example: 1/(3 + √2)
= 1/(3 + √2) × (3 − √2)/(3 − √2)
= (3 − √2) / (3² − (√2)²)
= (3 − √2) / (9 − 2)
= (3 − √2)/7.

This is the entire trick that the body of the lesson refers to — and once internalised, it makes any “1 over a binomial surd” question a 10-second job.

Why rationalisation matters

Three reasons:

1. Standard form. Examiners list answer choices in rationalised form. Even if your unrationalised expression is mathematically correct, it may not match any option until you rationalise.
2. Easier arithmetic. Adding or subtracting fractions with surd denominators forces you to find a common surd denominator — painful. After rationalisation, the denominator is a normal integer.
3. Comparison. Comparing 1/(√3 + 1) with 1/(√3 − 1) becomes obvious once you rationalise both — and is otherwise nearly impossible.

Real-world example

Indian Railways uses signalling cables whose impedance involves the square root of inductance-capacitance products. Every time an engineer divides one impedance by another, the answer comes back as a fraction with √(LC) in the denominator — and the standard practice is to rationalise so that the cable specification can be quoted as a rational decimal. The same arithmetic style appears in surveyor’s tables for railway track curvature, where chord lengths come out as 1/(a + √b) and have to be cleared up for site engineers.

For the RPF SI paper, the bridge between the conceptual idea and a quick solution is just this: see a surd in the denominator → call its conjugate → multiply on top and bottom → simplify.

Common misconception

Many candidates think rationalisation changes the value of the fraction. It does not. You are multiplying by something equal to 1 (conjugate over itself), so the value is preserved exactly. What changes is only the appearance of the fraction.

A second misconception is that the conjugate flips the sign of both terms. It does not — it flips only the sign between the terms. The conjugate of (3 + √2) is (3 − √2), not (−3 − √2). And the conjugate of (√5 − √3) is (√5 + √3), giving the product (√5)² − (√3)² = 5 − 3 = 2.

Worked example

Question: Simplify (4 + √3) / (4 − √3) and write the answer in the form p + q√3 where p, q are rational.

Solution:
Step 1: The denominator is the binomial surd (4 − √3). Its conjugate is (4 + √3).
Step 2: Multiply numerator and denominator by the conjugate:
(4 + √3)/(4 − √3) × (4 + √3)/(4 + √3) = (4 + √3)² / ((4 − √3)(4 + √3)).
Step 3: Denominator: 4² − (√3)² = 16 − 3 = 13.
Step 4: Numerator: (4 + √3)² = 16 + 8√3 + 3 = 19 + 8√3.
Step 5: Final expression: (19 + 8√3)/13 = 19/13 + (8/13)√3.
Conclusion: p = 19/13, q = 8/13.

A second quick example

Question: Simplify √75 − √48 + √27.

Solution:
Step 1: Break each radicand into a perfect-square factor times the rest:
√75 = √(25 × 3) = 5√3.
√48 = √(16 × 3) = 4√3.
√27 = √(9 × 3) = 3√3.
Step 2: All three are like surds (radicand 3), so they combine:
5√3 − 4√3 + 3√3 = (5 − 4 + 3)√3 = 4√3.
Conclusion: 4√3.

:::compare

Denominator form	Conjugate / multiplier	Resulting denominator
√a	√a	a
a + √b	a − √b	a² − b
a − √b	a + √b	a² − b
√a + √b	√a − √b	a − b
√a − √b	√a + √b	a − b
:::

:::keypoints

• A surd is an irrational root such as √2, √3 or ∛5.
• Surd laws: √a × √b = √(ab); √a / √b = √(a/b); (√a)² = a.
• Always simplify a surd by pulling out the largest perfect-square (or perfect-cube) factor.
• To rationalise a single-surd denominator, multiply top and bottom by that surd.
• For a binomial surd denominator, multiply by the conjugate so the product becomes a²−b.
• Rationalisation does not change the value — only the form — of a fraction.
• Like surds (same radicand) combine like algebraic terms; unlike surds do not.
• Mastering a handful of √-simplifications (√8, √12, …, √98) makes mental computation possible in the exam.
  :::

:::memory
“CSC” — three Cs to remember when you see a surd:

• Compress the surd to lowest form (pull out perfect squares).
• Spot the denominator form — single surd or binomial?
• Conjugate-multiply if binomial; same-surd-multiply if single.
  And: “flip the middle sign” to find a conjugate (a + √b → a − √b).
  :::

:::recap

• Simplify every surd to its lowest form before any arithmetic.
• For a single surd in the denominator, multiply top and bottom by that surd.
• For a binomial surd, multiply by the conjugate to get a² − b in the denominator.
• The value of the fraction never changes during rationalisation — only its presentation.
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Perfect-square unit digits can only be 0,1,4,5,6,9 — a number ending in 2,3,7,8 is never a perfect square. To find a 4-digit square root, split into pairs from the right; the tens digit comes from the larger pair, the units from the ending digit (5 is the giveaway since only 25 ends in 5). For cube roots of perfect cubes, the unit digit of the cube reveals the root's unit digit (cubes end: 1→1, 8→2, 7→3, 4→4, 5→5, 6→6, 3→7, 2→8, 9→9, 0→0). Example: cube root of 19683 — ends in 3 so unit is 7, and 27 (=3^3) < 19 < 64 so tens is 2, giving 27.