Summary of Index Laws
This subtopic brings the index laws together: multiplying, dividing, a power of a power and the zero index. In this NSW Year 9 Core maths subtopic each question uses two or three laws at once, so the order you apply them in matters. It is Stage 5 Core content under NESA outcome MA5‑IND‑C‑01.
You will decide which law to use first, clear brackets before multiplying or dividing, as in \(\frac{(3x^2)^2}{9x} = x^3\), and simplify the numerator and denominator separately before dividing. You will combine a power of a power with a zero index, work chains of divisions such as \(48y^8 \div 4y^3 \div 3y^2\), and judge whether someone else's quick simplification is right. These are the same steps used to simplify formulas in science. It is part of Year 9 Core Indices and Scientific Notation.
Every question with a fully worked solution.
- Summary of Index Laws - Video - Laws of Indices Watch
Theory
The index laws are four rules for simplifying powers with the same base. This Year 9 maths page puts them side by side and shows the order to use them: clear brackets, replace zero indices, then multiply and divide from left to right. The examples include coefficients, several letters, negative bases and fractions.
An index law is a shortcut for working with powers of the same base. There are four you need here.
Multiplication law: add the indices. \(x^4 \times x^3 = x^7\).
Division law: subtract the indices. \(x^9 \div x^2 = x^7\).
Power of a power: multiply the indices, and raise every factor in the bracket. \((3x^2)^3 = 27x^6\).
Zero index: anything except \(0\) to the power \(0\) is \(1\).
Numbers in front (coefficients) are worked out with normal arithmetic. With several letters, deal with each letter on its own.
For a base \(a\) (not \(0\)) and whole-number indices \(m\) and \(n\):
| When you see | Law | What to do |
|---|---|---|
| a bracket with an index | \((a^m)^n\) | raise every factor inside |
| a multiplication | \(a^m \times a^n\) | add the indices |
| a division or a fraction | \(a^m \div a^n\) | subtract the indices |
| a zero index | \(a^0\) | replace it by \(1\) |
How to simplify with several index laws
- Clear every bracket: raise each factor inside to the outside power.
- Replace any zero-index part by \(1\).
- Multiply: numbers together, then add the indices of each letter.
- Divide: numbers, then subtract the indices of each letter. In a fraction, finish the top and the bottom first. In a chain with no fraction line, multiply and divide from left to right.
- Check by substituting a small number.
(i) \(-6p^3q \times (-2pq^4)\)
(ii) \(7x^5 \times 3x^2 \div 21x^4\)
(i) Multiply the numbers. A negative times a negative is positive. Then add the indices of each letter. \(p\) on its own is \(p^1\).
| \(-6 \times (-2)\) | \(=\) | \(12\) |
| \(p^3 \times p^1\) | \(=\) | \(p^4\) |
| \(q^1 \times q^4\) | \(=\) | \(q^5\) |
| \(-6p^3q \times (-2pq^4)\) | \(=\) | \(\boldsymbol{12p^4q^5}\) |
(ii) Work from left to right. Multiply first, then divide.
| \(7x^5 \times 3x^2\) | \(=\) | \(21x^{5+2} = 21x^7\) |
| \(21x^7 \div 21x^4\) | \(=\) | \(1 \times x^{7-4}\) |
| \(\) | \(=\) | \(\boldsymbol{x^3}\) |
Check with \(x = 2\): \(224 \times 12 \div 336 = 8\), and \(2^3 = 8\).
(i) \((2^3)^2 \times 2^0 \div 2^4\)
(ii) \((-2)^5 \div (-2)^2 \times (-2)^0\)
(i) Bracket first: multiply the indices. Then add and subtract.
| \((2^3)^2 \times 2^0 \div 2^4\) | \(=\) | \(2^6 \times 2^0 \div 2^4\) |
| \(\) | \(=\) | \(2^{6+0-4}\) |
| \(\) | \(=\) | \(2^2 = \mathbf{4}\) |
(ii) The bracket makes \(-2\) the base. Subtract, then add the indices.
| \((-2)^5 \div (-2)^2 \times (-2)^0\) | \(=\) | \((-2)^{5-2+0}\) |
| \(\) | \(=\) | \((-2)^3\) |
| \(\) | \(=\) | \(\mathbf{-8}\) |
The index \(3\) is odd, so three negatives multiplied give a negative answer.
Clear the bracket first. All three factors inside are cubed.
| \((2m^2n)^3\) | \(=\) | \(2^3 \times m^{2 \times 3} \times n^3\) |
| \(\) | \(=\) | \(8m^6n^3\) |
Now multiply. Numbers first, then each letter on its own.
| \(8 \times 3\) | \(=\) | \(24\) |
| \(m^6 \times m^1\) | \(=\) | \(m^7\) |
| \(n^3 \times n^2\) | \(=\) | \(n^5\) |
| \((2m^2n)^3 \times 3mn^2\) | \(=\) | \(\boldsymbol{24m^7n^5}\) |
Finish the top first. Square every factor in the bracket.
| \((6a^4b^2)^2\) | \(=\) | \(6^2 \times a^{4 \times 2} \times b^{2 \times 2}\) |
| \(\) | \(=\) | \(36a^8b^4\) |
Now divide, one piece at a time.
| \(36 \div 4\) | \(=\) | \(9\) |
| \(a^8 \div a^3\) | \(=\) | \(a^5\) |
| \(b^4 \div b^1\) | \(=\) | \(b^3\) |
| \(\dfrac{(6a^4b^2)^2}{4a^3b}\) | \(=\) | \(\boldsymbol{9a^5b^3}\) |
Cancelling before the bracket is cleared is where most mistakes happen.
Common pitfalls
Frequently asked questions
What are the index laws?
Multiply powers with the same base: add the indices. Divide: subtract them. A power of a power: multiply them. And \(a^0 = 1\) for any \(a\) except \(0\).
Which index law do you use first?
Clear the brackets first, then replace any zero index by \(1\), then multiply and divide from left to right. In a fraction, simplify the top and the bottom before dividing.
Can you add indices when you add terms?
No. \(x^2 + x^5\) cannot be simplified. The index laws only work for multiplying and dividing powers with the same base.
How do you simplify index laws with numbers in front?
Work the numbers out with normal arithmetic, then use the index laws on each letter: \(3x^2 \times 5x^4 = 15x^6\).
How can I check my index law answer?
Substitute a small number such as \(2\) into the question and into your answer. If the two values are different, there is a mistake.