Surds & Indices: Complete Learning Suite

Zero to Hero 🚀

Surds & Indices Complete Workshop

Detailed Theory + Step-by-Step Solutions

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The Rule Book (Read First)

A. Laws of Indices (The Power Rules)

1. Multiplication Rule

If bases are same, ADD the powers.

$$ a^m \times a^n = a^{m+n} $$

Ex: $2^3 \times 2^4 = 2^{3+4} = 2^7$

2. Division Rule

If bases are same, SUBTRACT the powers.

$$ \frac{a^m}{a^n} = a^{m-n} $$

Ex: $5^6 \div 5^2 = 5^{6-2} = 5^4$

3. Power of Power

MULTIPLY the powers.

$$ (a^m)^n = a^{m \times n} $$

Ex: $(3^2)^3 = 3^{2 \times 3} = 3^6$

4. The Zero & Negative Rule

Anything power 0 is 1. Negative power flips the fraction.

$$ a^0 = 1 \quad | \quad a^{-n} = \frac{1}{a^n} $$

Ex: $99^0 = 1$ | $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$

B. Laws of Surds (The Root Rules)

A Surd is an irrational root (like $\sqrt{2}, \sqrt{3}, \sqrt{5}$). $\sqrt{4}$ is NOT a surd because it equals 2.

Addition (Only Like Terms) $$ 2\sqrt{3} + 5\sqrt{3} = 7\sqrt{3} $$ $\sqrt{2} + \sqrt{3} \neq \sqrt{5}$ (Never!)
Multiplication $$ \sqrt{a} \times \sqrt{b} = \sqrt{ab} $$ Ex: $\sqrt{2} \times \sqrt{3} = \sqrt{6}$
Rationalization To remove root from bottom: $$ \frac{1}{\sqrt{a}} \times \frac{\sqrt{a}}{\sqrt{a}} = \frac{\sqrt{a}}{a} $$

C. The Toolkit: Formulas & Values

KEY IDENTITIES

🔐 Algebraic Identities

  • $(a+b)^2$ $a^2 + 2ab + b^2$
  • $(a-b)^2$ $a^2 – 2ab + b^2$
  • $a^2 – b^2$ $(a+b)(a-b)$
  • $(a+b)^3$ $a^3 + b^3 + 3ab(a+b)$
  • $(a-b)^3$ $a^3 – b^3 – 3ab(a-b)$
  • $(a+b+c)^2$ $a^2+b^2+c^2+2(ab+bc+ca)$
CHEAT SHEET

💎 Critical Surd Values

Memorize these up to 3 decimal places for speed.

Square Root of 2 $\sqrt{2} \approx 1.414$
Square Root of 3 $\sqrt{3} \approx 1.732$
Square Root of 5 $\sqrt{5} \approx 2.236$
Square Root of 10 $\sqrt{10} \approx 3.162$
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Practice Worksheet

LEVEL 1: ROOKIE (Class 5-7)
LEVEL 2: CHALLENGER (Class 8-9)
LEVEL 3: MASTER (Class 10)
LEVEL 4: LEGEND (Pro Level)
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THE TITAN GAUNTLET

50 Pro-Level Questions for Olympiads & Competitive Exams

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