Page 2: Indices - Powers and Roots

This page delves into the concept of indices, which is crucial for understanding how to estimate in maths and perform complex calculations efficiently.

The guide begins by explaining the basics of square numbers and cubes, providing examples such as 2² = 2 × 2 = 4 and 2³ = 2 × 2 × 2 = 8. It then progresses to higher powers, demonstrating how to calculate 2⁴ and 2⁵.

Definition: An index (plural: indices) is the power to which a number is raised.

The page introduces negative indices, explaining that they represent the reciprocal of the positive power. For instance, 2⁻² = 1/2².

Example: 2⁻³ = 1/2³ = 1/8

Fractional indices are also covered, with a focus on square roots and cube roots. The guide explains that a½ is equivalent to √a, and a⅓ is the cube root of a.

Highlight: Any number raised to the power of 0 equals 1, and any number raised to the power of 1 equals itself.

The page concludes with rules for multiplying and dividing indices with the same base, which are essential for estimating calculations in Maths Genie and similar resources.

Example: When multiplying indices with the same base, add the powers: 2³ × 2⁴ = 2³⁺⁴ = 2⁷

Page 4: LCM and HCF - Number Theory Fundamentals

The final page focuses on Lowest Common Multiple (LCM) and Highest Common Factor (HCF), which are essential concepts for understanding HCF and LCM in Edexcel exams.

The guide provides step-by-step methods for finding the LCM and HCF of two or more numbers using prime factorization.

Definition: LCM (Lowest Common Multiple) is the smallest positive number that is divisible by two or more numbers.

For finding the LCM, the method involves:

1. Breaking down each number into its prime factors
2. Taking each prime factor to the highest power in which it occurs in either number
3. Multiplying these factors together

Example: To find the LCM of 180 and 220: 180 = 2² × 3² × 5 220 = 2² × 5 × 11 LCM = 2² × 3² × 5 × 11 = 1980

The HCF calculation is explained as finding the product of all common prime factors:

Definition: HCF (Highest Common Factor) is the largest positive integer that divides each of the numbers without a remainder.

Example: To find the HCF of 24 and 36: 24 = 2³ × 3 36 = 2² × 3² HCF = 2² × 3 = 12

This page is crucial for students tackling HCF and LCM GCSE questions and answers, providing a solid foundation for more advanced number theory concepts.