How to teach exponent laws and irrational numbers
Year 7 to Year 9 (ages 12 to 15)
Exponent (index) laws are shortcuts for multiplying, dividing and raising powers with the same base. Irrational numbers are numbers, like most square roots and pi, that cannot be written as an exact fraction and whose decimals never terminate or repeat.
How to teach it
- Start with the multiplication law: a^m x a^n = a^(m+n), showing why by writing out the repeated multiplication in full for small numbers.
- Introduce the division law (subtract exponents) and the power-of-a-power law (multiply exponents) the same way, always checking with a full expansion first.
- Teach square numbers and roots before irrational numbers, so 'rational' (a perfect square's root) has a clear contrast with 'irrational' (any other root).
- Classify a mix of numbers (fractions, decimals, square roots, pi) as rational or irrational, justifying each with the fraction/decimal test.
- Keep base numbers small and positive until the three exponent laws are automatic, then introduce zero and negative exponents.
Worked example
Simplify 3^4 x 3^2 Add the exponents (same base): 3^(4+2) = 3^6 Is sqrt(9) rational or irrational? sqrt(9) = 3, a whole number, so rational Is sqrt(7) rational or irrational? 7 is not a perfect square, so irrational
Common mistakes
- Multiplying the exponents instead of adding them for a^m x a^n (that rule is for (a^m)^n).
- Assuming every square root is irrational, forgetting perfect squares like sqrt(16) = 4 are whole numbers.
- Believing a long decimal is automatically irrational, when a long REPEATING decimal is still rational.
- Adding the bases instead of keeping the base the same and combining only the exponents.
Frequently asked questions
What are the exponent (index) laws?
When multiplying powers with the same base, add the exponents: a^m x a^n = a^(m+n). When dividing, subtract: a^m / a^n = a^(m-n). When raising a power to a power, multiply: (a^m)^n = a^(m x n).
What is an irrational number?
A number that cannot be written as an exact fraction of two integers. Its decimal expansion never terminates and never repeats. Square roots of non-perfect squares (like sqrt(2)) and pi are the most common examples.
How do you know if a number is rational or irrational?
If it can be written as a fraction, or its decimal terminates or repeats in a pattern, it is rational. Square roots of perfect squares (4, 9, 16...) are rational; square roots of everything else, and pi, are irrational.
What year are exponent laws and irrational numbers taught?
In the Australian Curriculum, primes and exponent notation start at Year 7 (AC9M7N02), irrational numbers and exponent laws at Year 8 (AC9M8N01-02), and the real number system (rational and irrational together) at Year 9 (AC9M9N01, AC9M9A01).
Practise with free worksheets
Printable worksheets with answer keys that are never wrong.