Structure and calculation Revision Notes

Powers

For a positive integer n, a^n means n factors of a multiplied together. Useful common powers include 2^2 = 4, 3^2 = 9, 4^2 = 16, 5^2 = 25, 10^2 = 100, 2^3 = 8, 3^3 = 27 and 5^3 = 125.

Roots

A square root reverses squaring, a cube root reverses cubing, and higher roots reverse the matching power. Check a root by raising the proposed answer to the stated power; for example, 4^4 = 256, so the fourth root of 256 is 4.

Calculation checks

When working with powers, write repeated multiplication if the value is not immediately familiar. For a root, identify the positive number whose corresponding power gives the number inside the root, then substitute it back to verify the result.

Key vocabulary

A factor divides into a number with no remainder. A multiple is a result of multiplying a given number by an integer. A prime has exactly two positive factors, whereas a composite number has more than two. The number 1 is neither prime nor composite.

Prime factorisation and notation

To find a prime factorisation, divide repeatedly by prime numbers until only primes remain. Write repeated factors compactly with powers: 72 = 2 × 2 × 2 × 3 × 3 = 2^3 × 3^2. The order of the prime factors does not change the factorisation.

HCF and LCM

For prime factorizations, the HCF is formed from prime factors shared by all numbers, taking the smallest power available. The LCM contains each prime factor appearing in any number, taking the largest power available. Check an HCF by confirming it divides every number, and check an LCM by confirming every number divides it.

Simplifying and combining roots

Rewrite a square root by taking out the largest square factor. For example, √50 = √(25 × 2) = 5√2. After simplifying, add or subtract only like surds: 3√5 + 2√5 = 5√5, whereas √2 and √3 cannot be combined.

Laws of integer indices

For a non-zero base a, a^m × a^n = a^(m+n), a^m ÷ a^n = a^(m−n), and (a^m)^n = a^(mn). Also, a^0 = 1 and a^(−n) = 1/a^n. Apply one law at a time and simplify the resulting power.

How to organise a list

Choose one part of the situation to keep fixed while varying another part. For example, fix the first digit and list every allowed second digit before changing the first digit. This creates clear groups and makes the list easier to check.

Checking completeness

After listing, count the outcomes expected from the choices available and compare this with the number written down. Check that every outcome follows the rules, that no outcome is repeated, and that each possible starting choice has been considered.

Useful representations

Possibilities may be shown in rows, a table, a tree diagram or an ordered list. The representation is effective when its structure shows how the outcomes were generated and makes omissions or repetitions easy to identify.