Surds, Indices & Square/Cube Roots

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.

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.

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.

Cover the answer, recall, then check. 12 cards on laws of indices, surds and roots for RPF SI.

Q1. Product law of indices?
A1. a^m × a^n = a^(m+n). E.g. 2³×2⁴ = 2⁷ = 128.

Q2. Quotient law?
A2. a^m ÷ a^n = a^(m−n). E.g. 5⁶ ÷ 5⁴ = 5² = 25.

Q3. Power of a power?
A3. (a^m)^n = a^(mn). E.g. (2³)² = 2⁶ = 64.

Q4. Zero and negative index?
A4. a⁰ = 1 (a≠0); a^(−n) = 1/a^n.

Q5. Meaning of a^(m/n)?
A5. The n-th root of a^m, i.e. (ⁿ√a)^m. E.g. 8^(2/3) = (∛8)² = 4.

Q6. Rationalise 1/(√5 − √3).
A6. Multiply by (√5+√3): (√5+√3)/(5−3) = (√5+√3)/2.

Q7. √(a×b) equals?
A7. √a × √b (for a,b ≥ 0). E.g. √(4×9) = 2×3 = 6.

Q8. Square root of 1764?
A8. 42 (since 42² = 1764).

Q9. Cube root of 3375?
A9. 15 (since 15³ = 3375).

Q10. Unit digit of a perfect square — which digits are impossible?
A10. 2, 3, 7, 8 can never be the unit digit of a perfect square.

Q11. Simplify √50 + √18.
A11. 5√2 + 3√2 = 8√2.

Q12. Value of (√3 + √2)(√3 − √2)?
A12. (√3)² − (√2)² = 3 − 2 = 1.

Worked Example

Problem: If 2ˣ = 8^(y+1) and 9ʸ = 3^(x−9), find the values of x and y.

Solution:
Step 1 — Write both sides of each equation with the same base.
Equation 1: 2ˣ = 8^(y+1). Since 8 = 2³, we get 8^(y+1) = 2^(3(y+1)) = 2^(3y+3).
Equal bases ⇒ equal powers: x = 3y + 3. … (i)
Step 2 — Equation 2: 9ʸ = 3^(x−9). Since 9 = 3², we get 9ʸ = 3^(2y).
Equal bases ⇒ 2y = x − 9, so x = 2y + 9. … (ii)
Step 3 — Solve (i) and (ii) together. From (i) and (ii): 3y + 3 = 2y + 9 → y = 6.
Step 4 — Substitute back: x = 3(6) + 3 = 21. Check with (ii): x = 2(6) + 9 = 21. ✓

Answer: x = 21 and y = 6.