Formula Sheet • Powers, Surds & Prime Factors

Numbers & Indices Formula Sheet

All essential number theory and exponent formulas in one place — index laws, fractional roots, standard form arithmetic, surd simplification and rationalising denominators, HCF, and LCM.

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A4 / Letter portrait optimized • Pure black-and-white ink saving mode Mode: Comprehensive Study Guide (With Examples)

The Six Index Laws

Product Rule (Multiply Same Base)

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

When multiplying powers with the identical base, add the exponents.

Worked Example

\displaystyle 2^3 \times 2^4 = 2^{3+4} = 2^7 = 128

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Quotient Rule (Divide Same Base)

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

When dividing powers with the identical base, subtract the exponents.

Worked Example

\displaystyle 5^6 \div 5^2 = 5^{6-2} = 5^4 = 625

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Power of a Power

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

When raising a power to another power, multiply the exponents together.

Worked Example

\displaystyle (3^2)^4 = 3^{2 \times 4} = 3^8 = 6561

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Zero Exponent

a^0 = 1 \quad (a \neq 0)

Any non-zero real number raised to the zero power evaluates strictly to 1.

Worked Example

\displaystyle 7^0 = 1, \quad (-42)^0 = 1, \quad 10\,000^0 = 1

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Negative Exponent

a^{-n} = \frac{1}{a^n}

A negative exponent represents the multiplicative reciprocal of the positive power.

Worked Example

\displaystyle 2^{-3} = \frac{1}{2^3} = \frac{1}{8}, \quad \left(\frac{2}{3}\right)^{-2} = \frac{9}{4}

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Fractional Exponent (Roots)

a^{\frac{m}{n}} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m

Denominator n denotes the root index; numerator m denotes the power.

Worked Example

\displaystyle 8^{\frac{2}{3}} = (\sqrt[3]{8})^2 = 2^2 = 4

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Standard Form (Scientific Notation)

Standard Form Definition

A \times 10^n \quad (1 \le A < 10, \; n \in \mathbb{Z}

A must be between 1 and 10 (1 ≤ A < 10) • n is an integer.

Worked Example

\displaystyle 3\,450\,000 = 3.45 \times 10^6, \quad 0.00072 = 7.2 \times 10^{-4}

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Multiplying in Standard Form

(a \times 10^m) \times (b \times 10^n) = (ab) \times 10^{m+n}

Multiply coefficients together and add exponents of 10. Re-adjust if ab ≥ 10.

Worked Example

\displaystyle (4 \times 10^4) \times (5 \times 10^3) = 20 \times 10^7 = 2.0 \times 10^8

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Dividing in Standard Form

\frac{a \times 10^m}{b \times 10^n} = \left(\frac{a}{b}\right) \times 10^{m-n}

Divide coefficients and subtract exponents of 10. Re-adjust if a/b < 1.

Worked Example

\displaystyle \frac{9 \times 10^6}{3 \times 10^2} = \left(\frac{9}{3}\right) \times 10^{6-2} = 3 \times 10^4

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Surds & Radicals

Product Rule of Surds

\sqrt{ab} = \sqrt{a} \times \sqrt{b}

Factor out the largest perfect square (4, 9, 16, 25, …) to simplify.

Worked Example

\displaystyle \sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2} = 6\sqrt{2}

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Quotient Rule of Surds

\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}

Splits a fractional root into independent numerator and denominator roots.

Worked Example

\displaystyle \sqrt{\frac{25}{49}} = \frac{\sqrt{25}}{\sqrt{49}} = \frac{5}{7}

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Adding & Subtracting Like Surds

p\sqrt{k} \pm q\sqrt{k} = (p \pm q)\sqrt{k}

Only surds with the identical number under the radical can be combined.

Worked Example

\displaystyle 5\sqrt{3} + 4\sqrt{3} - 2\sqrt{3} = 7\sqrt{3}

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Difference of Squares (Surds)

(\sqrt{a} + \sqrt{b})(\sqrt{a} - \sqrt{b}) = a - b

Multiplying conjugate surd pairs cancels middle terms and yields a rational integer.

Worked Example

\displaystyle (\sqrt{5} + \sqrt{2})(\sqrt{5} - \sqrt{2}) = 5 - 2 = 3

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Rationalising Single Denominator

\frac{a}{\sqrt{b}} = \frac{a\sqrt{b}}{b}

Multiply top and bottom by √b so the denominator becomes an integer.

Worked Example

\displaystyle \frac{6}{\sqrt{3}} = \frac{6\sqrt{3}}{3} = 2\sqrt{3}

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Rationalising Conjugate Denominator

\frac{c}{a + \sqrt{b}} = \frac{c(a - \sqrt{b})}{a^2 - b}

Multiply top and bottom by the conjugate (a − √b) to eliminate the root.

Worked Example

\displaystyle \frac{4}{3 + \sqrt{5}} = \frac{4(3 - \sqrt{5})}{9 - 5} = 3 - \sqrt{5}

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Factors, Primes & Multiples

Prime Factorisation

N = 2^a \times 3^b \times 5^c \times \dots

Every integer greater than 1 uniquely decomposes into a product of prime numbers.

Worked Example

\displaystyle 360 = 2^3 \times 3^2 \times 5

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Highest Common Factor (HCF)

\text{HCF}(a, b) = \frac{a \times b}{\text{LCM}(a, b)}

Multiply the lowest powers of all shared prime factors, or use HCF × LCM = a × b.

Worked Example

\displaystyle \text{HCF}(24, 36) = 2^2 \times 3^1 = 12

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Lowest Common Multiple (LCM)

\text{LCM}(a, b) = \frac{a \times b}{\text{HCF}(a, b)}

Multiply the highest powers of all prime factors appearing in either number.

Worked Example

\displaystyle \text{LCM}(24, 36) = \frac{24 \times 36}{12} = 72

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Curriculum & Revision Note

About These Numbers & Indices Formulas

Number theory and index laws form the computational backbone for all secondary school and STEM coursework. This sheet covers the six core laws of exponents (including fractional and negative indices), scientific standard form notation, surd operations and conjugate rationalisation, alongside prime factorisation methods for HCF and LCM.

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