This second course on exponents takes the laws you already know into new ground. It sets out fourteen laws, from multiplying powers to roots written as fractional exponents, and uses them to evaluate expressions such as 25 to the power 3/2 without a calculator. You will simplify and prove results with letters in the exponent, many of which collapse to 1, solve exponential equations by bringing both sides to the same base, and handle chains like 2ˣ = 3ʸ = 12ᶻ with the k method. The last two lessons are mixed problems: decimal bases and exponents, surds, hidden identities, negative bases and nested powers.
Middle Years
Exponents (Level 2)
The fourteen laws of exponents, with negative and fractional exponents, letters in the exponent, exponential equations, chains of equal powers, and mixed problems with decimals, surds and identities.
- 7 lessons
Overview
Lessons
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The laws of exponents
All fourteen laws in one place: products, quotients and powers of powers, zero and negative exponents, negative bases, equal powers, and roots as fractional exponents.
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Evaluating expressions with exponents
Write every base as a power of a prime, multiply out the exponents and collect each prime, to evaluate expressions with fractional and negative exponents.
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Letters in the exponent
Simplify and prove results such as (xᵃ/xᵇ)ᵃ⁺ᵇ (xᵇ/xᶜ)ᵇ⁺ᶜ (xᶜ/xᵃ)ᶜ⁺ᵃ = 1, using cyclic patterns, a + b + c = 0 and the sum of cubes.
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Exponential equations
Solve equations with the unknown in the exponent by bringing both sides to the same base and equating exponents, including two bases and repeated powers.
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Equal powers and the k method
Prove relations such as 1/z = 1/y + 2/x when 2ˣ = 3ʸ = 12ᶻ by setting each power equal to k, writing every base as a power of k and comparing exponents.
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Decimals, roots and identities with exponents
Short problems with decimal bases and exponents, surds inside the base, powers of 10, and algebraic identities hidden behind fractional powers.
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Negative bases, nested powers and mixed problems
Negative fractional exponents on fractions, a negative base under a cube root, powers nested inside powers, and equations whose answer is used to find another value.
Practice papers
Ten question papers that follow the lessons in order. Each has 20 questions for 50 marks (1 hour 30 minutes), with a key sheet and worked solutions.
MCQ tests
Ten tests of 25 multiple-choice questions (30 minutes each). Answer them on the page and check your score, or download the PDF with its key and solutions.