Multiplication and Division Index Laws
Multiplication and Division Index Laws in NSW Year 10 Core maths revises and extends the two main index laws (NESA outcome MA5‑IND‑C‑01). When powers with the same base are multiplied, the indices are added; when they are divided, the indices are subtracted, so \(k^9 \div k^4 = k^5\).
You will simplify numerical powers such as \(5^3 \times 5^6 \div 5^2\), multiply and divide algebraic terms with coefficients and several pronumerals, and find a missing index or a missing term. You will also write expressions for the area of a poster, the length of a rectangle and the volume of a box whose dimensions are algebraic terms. It is part of Year 10 Core Indices and Surds.
Theory
The index laws are shortcuts for powers of the same base. To multiply, add the indices; to divide, subtract them. This Year 10 maths page applies the laws to numbers and to algebraic terms with coefficients, and to finding a missing index or term.
In \(a^n\), \(a\) is the base and \(n\) is the index (or power, or exponent). \(a^n\) means \(n\) factors of \(a\) multiplied together: \(a^4 = a \times a \times a \times a\). A pronumeral on its own has index \(1\): \(a = a^1\).
The multiplication law: when multiplying powers of the same base, keep the base and add the indices, because you are counting all the factors.
The division law: when dividing powers of the same base, keep the base and subtract the indices, because the matching factors cancel.
With coefficients (the numbers in front), multiply or divide the numbers as usual, then apply the laws to each pronumeral separately. A coefficient that does not divide exactly stays as a fraction.
The laws only work for the same base. \(2^3 \times 5^2\) cannot be combined into a single power.
For any base \(a\) and positive integers \(m\) and \(n\):
With coefficients:
How to simplify with the index laws
- Check the bases are the same. Write any lone pronumeral as power \(1\).
- Deal with the numbers: multiply them, or divide them as a fraction in simplest form.
- Add the indices of each pronumeral when multiplying; subtract when dividing.
- Write the answer with the number first, then the pronumerals in alphabetical order.
Finding a missing index
- Simplify one side to a single power of the base.
- Match the indices on both sides and solve the equation.
Work from left to right. Add the indices to multiply, subtract to divide.
| \(7^5 \times 7^2 \div 7^4\) | \(=\) | \(7^{5 + 2 - 4}\) |
| \(\) | \(=\) | \(7^3\) |
| \(\) | \(=\) | \(\boldsymbol{343}\) |
Simplifying first keeps the numbers small: \(7^5 = 16\,807\) is never needed.
Multiply the numbers, then add the indices of each pronumeral. \(a\) means \(a^1\).
| \(-4a^3b \times 5ab^2\) | \(=\) | \((-4 \times 5) \times a^{3 + 1} \times b^{1 + 2}\) |
| \(\) | \(=\) | \(\boldsymbol{-20a^4b^3}\) |
Simplify the numbers as a fraction, using their HCF, \(6\). Subtract the indices of each pronumeral.
| \(\dfrac{12x^6y^3}{18x^2y}\) | \(=\) | \(\dfrac{12}{18} \times x^{6 - 2} \times y^{3 - 1}\) |
| \(\) | \(=\) | \(\dfrac{2}{3}x^4y^2\) |
| \(\) | \(=\) | \(\boldsymbol{\dfrac{2x^4y^2}{3}}\) |
The \(\dfrac{2}{3}\) stays as a fraction; it is not \(0.67\).
(i) Find \(n\) if \(3^n \times 3^4 = 3^{10}\).
(ii) Find the missing term: \(8y^3 \times \square = 56y^{10}\).
(i) The bases match, so the indices must match: \(n + 4 = 10\).
| \(n + 4\) | \(=\) | \(10\) |
| \(\therefore\ n\) | \(=\) | \(\boldsymbol{6}\) |
(ii) The missing term is the answer divided by the known term.
| \(\square\) | \(=\) | \(\dfrac{56y^{10}}{8y^3}\) |
| \(\) | \(=\) | \(\boldsymbol{7y^7}\) |
Check: \(8y^3 \times 7y^7 = 56y^{10}\).
Common pitfalls
Frequently asked questions
What are the index laws for multiplying and dividing?
To multiply powers of the same base, add the indices: \(a^m \times a^n = a^{m + n}\). To divide, subtract them: \(a^m \div a^n = a^{m - n}\).
Why do you add indices when multiplying?
Because an index counts factors. \(a^3 \times a^2\) is three factors of \(a\) times two more, which is five factors of \(a\) altogether: \(a^5\).
What do you do with the numbers in front?
Multiply or divide them as normal numbers. The index laws only apply to the indices. For example, \(-4a^3 \times 5a = -20a^4\).
Can you use the index laws with different bases?
Not directly. \(2^3 \times 3^2\) cannot be written as one power. The bases must be the same before you add or subtract the indices.
What is x to the power 1?
It is just \(x\). Any pronumeral written without an index has index \(1\), which matters when you add indices: \(x^2 \times x = x^3\).