GCSE Maths · Standard Form, Indices and Surds · Higher · Summary

GCSE Maths Standard Form, Indices and Surds (Higher): summary and key words

Part of the GCSE Maths standard form, indices and surds chapter (Higher). Chapter overview · Foundation version of this lesson · whole chapter on one page

Chapter summary

Skill	Method
Index laws (same base)	Multiply: add indices. Divide: subtract indices. Power of a power: multiply indices. a¹ = a.
Zero and negative indices	a⁰ = 1. a⁻ⁿ = 1/aⁿ. For fractions, flip the fraction and make the index positive.
Indices with letters	Deal with the number and each letter separately. (2a³)⁴ = 16a¹².
Roots	√a: number that multiplies by itself to give a. Between two whole numbers: use the perfect squares or cubes on either side. x² = k has two solutions.
Fractional indices (Higher)	a^m/n = (n-th root of a)^m. Do the root first. Negative: reciprocal first. Solve x^p/q = k by raising to the power q/p.
Standard form	A × 10ⁿ with 1 ≤ A < 10. Big number: positive n. Number less than 1: negative n. Count the places the decimal point moves.
Calculating in standard form	Multiply: multiply the A values, add the powers. Divide: divide the A values, subtract the powers. Add or subtract: make the powers match. Then tidy so that 1 ≤ A < 10.
Comparing standard form	Compare the powers of 10 first; if equal compare the front numbers.
Exact answers with π	Leave π in the answer: area 36π. Treat π like a letter. Use the radius for area.
Simplifying surds (Higher)	√a × √b = √(ab). Use the largest square factor: √12 = 2√3. Add or subtract only like surds.
Expanding with surds (Higher)	Every term × every term. √a × √a = a. (a + √b)(a − √b) = a² − b.
Rationalising (Higher)	k/(c√n): multiply top and bottom by √n. k/(a + √b): multiply top and bottom by the conjugate (a − √b).

Can you…?

I can write numbers using index notation and use the three index laws with numbers and letters.
I know that a⁰ = 1 and that a negative index means “one over”.
I can work out roots, place a root between two whole numbers, and use a calculator for powers and roots.
I can write any large or small number in standard form and convert back.
I can compare, order and calculate with numbers in standard form, with and without a calculator.
I can give exact answers for circle questions in terms of π.
(Higher) I can work with fractional indices and solve index equations.
(Higher) I can simplify, add, multiply and expand surds.
(Higher) I can rationalise a denominator, including using the conjugate.

Key terms

Base: The number that is multiplied in a power. In 3⁴ the base is 3.
Index (power, exponent): The small raised number that says how many lots of the base are multiplied. Plural: indices.
Reciprocal: One divided by a number. The reciprocal of 5 is 1/5 and a⁻¹ = 1/a.
Root: The number that gives the original when raised to a power. √25 = 5 and ³√8 = 2.
Perfect square / cube: The square or cube of a whole number: 1, 4, 9, 16… and 1, 8, 27, 64…
Standard form: A number written as A × 10ⁿ with 1 ≤ A < 10 and n a whole number.
Significant figures: The digits of a number counted from the first non-zero digit.
Exact answer: An answer that has not been rounded, such as 36π or 2√3.
Surd: A square root that is not a whole number or fraction, such as √2. It is irrational.
Irrational number: A number that cannot be written as a fraction. Its decimal never ends or repeats.
Like surds: Surds with the same number under the root, for example 3√2 and 5√2.
Rationalise the denominator: Rewrite a fraction so that its bottom is a whole number, with no surd.
Conjugate: For a + √b, the conjugate is a − √b (the sign in the middle changed). Their product is a² − b.