Surds, Indices and Exponents
The laws of indices, fractional and negative powers as roots and reciprocals, simplifying and comparing surds, rationalising a denominator, and solving simple exponential equations by matching bases.
What you will be able to do
- Apply the laws of indices for products, quotients and powers of powers.
- Read a fractional index as a root and a negative index as a reciprocal.
- Simplify a surd, rationalise a denominator and compare two surds.
- Solve an exponential equation by writing both sides to a common base.
What is inside this lesson
- 1What an index is
- 2The laws of indices
- 3Fractional and negative indices
- 4Simplifying and comparing surds
- 5Rationalising a denominator
- 6Exponential equations by matching bases
- 7Worked examples
- 8Common mistakes
- 9Quick revision
- 10Practice
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What an index is
An index, or exponent, is the small number that says how many times a base is multiplied by itself: in 2^5 the base is 2 and the index is 5, so the value is 32.
Exam questions in this area are quick if you know the laws below and slow if you do not. They come as a chain of powers to simplify, a fractional power to evaluate, a surd expression to tidy or compare, or an equation like 4 to the power (x plus 1) equals 8 to the power x. All of these reduce to writing everything with the same base.
The caret sign here means "to the power of".
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