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Rational Exponents

Maths: Edexcel A Level Pure Maths, topic 2.1.2

Rational Exponents

Rational exponents are exponents that are fractions, where the numerator is a power and the denominator is a root.

Notation

Notation

  • There are multiple ways of writing an expression, a variable, or a number with a rational exponent:
  • amn=(a1n)m=(am)1n=amn=(a)ma^{\frac{m}{n}} = (a^{\frac{1}{n}})^{m} = (a^m)^{\frac{1}{n}}=\sqrt[n]{a^m} = (\sqrt{a})^m
Solving rational exponents

Solving rational exponents

  • To solve rational equations we raise both sides of the equation to the reciprocal of the exponent.
  • This eliminates the exponent on the variable term as any number multiplied by its reciprocal equals 1.
  • This is the same as square rooting both sides of x2=4x^2 = 4 to get x=±2x = \pm 2.
Example

Example

  • We want to solve the equation x54=32x^{\frac{5}{4}} = 32.
Raise to the reciprocal exponent

Raise to the reciprocal exponent

Take the fifth root of 32

Take the fifth root of 32

Raise two to the power four

Raise two to the power four

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