Index Laws for Multiplying and Dividing
Theory
When multiplying terms with the same base, the indices are added: \(a^m \times a^n = a^{m+n}\). When dividing, they are subtracted: \(a^m \div a^n = a^{m-n}\). The base can be a number or a pronumeral. This Year 8 maths page shows both laws with coefficients.
Multiplying: \(a^m \times a^n = a^{m+n}\). Keep the base, add the indices. Dividing: \(a^m \div a^n = a^{m-n}\). Keep the base, subtract the indices. Both laws need the same base.
A number or pronumeral on its own has index \(1\): \(p = p^1\), so \(p^6 \div p = p^5\).
A number in front is separate. Multiply or divide the coefficients as ordinary numbers, then use the law on the powers: \(3x^4 \times 2x^3 = (3 \times 2) \times x^{4+3} = 6x^7\), and \((5 \times 7^3) \times (2 \times 7^6) = 10 \times 7^9\). If the numbers do not divide exactly, leave a fraction.
Two different bases stay apart: group like bases and work on each one. When the bottom index is bigger, the leftover power stays on the bottom: \(\dfrac{11^2}{11^5} = \dfrac{1}{11^3}\).
Powers of 10. Large amounts written as a number times a power of \(10\) can be multiplied and divided with the laws in one step.
The index laws for multiplying and dividing:
How to simplify a product or quotient of powers
- Group the numbers in front, and each base on its own.
- Multiply or divide the numbers in front.
- Add indices when multiplying, subtract when dividing.
| \(3x^4 \times 2x^3\) | \(=\) | \((3 \times 2) \times x^4 \times x^3\) |
| \(\) | \(=\) | \(6x^{4+3} = 6x^7\) |
\(6x^7\)
| \(12p^6 \div 4p\) | \(=\) | \((12 \div 4) \times p^6 \div p^1\) |
| \(\) | \(=\) | \(3p^{6-1} = 3p^5\) |
\(3p^5\)
(i) \(4^5 \times 4^3\)
(ii) \(12^7 \div 12^3\)
| \(4^5 \times 4^3\) | \(=\) | \(4^8\) |
| \(12^7 \div 12^3\) | \(=\) | \(12^4\) |
(i) \(4^8\)
(ii) \(12^4\)
(i) \((6 \times 2^5) \times (4 \times 2^3)\)
(ii) \((30 \times 8^9) \div (5 \times 8^6)\)
| \(\) | \(=\) | \(24 \times 2^8\) |
| \(\) | \(=\) | \(6 \times 8^3 = 3072\) |
(i) \(24 \times 2^8\)
(ii) \(6 \times 8^3 = 3072\)
(i) \(\dfrac{30a^7b^5}{6a^3b}\)
(ii) \(4m^3n \times 5m^2n^4 \div 10mn^3\)
| \(\dfrac{30a^7b^5}{6a^3b}\) | \(=\) | \(5a^4b^4\) |
| \(20m^5n^5 \div 10mn^3\) | \(=\) | \(2m^4n^2\) |
(i) \(5a^4b^4\)
(ii) Multiply first (left to right): \(2m^4n^2\)
(ii) A plan gives \(8 \times 10^{10}\) bytes of data. A song uses \(4 \times 10^6\) bytes. How many songs is that?
| \({?} + 5\) | \(=\) | \(12\) |
| \((8 \div 4) \times 10^{10-6}\) | \(=\) | \(2 \times 10^4\) |
(i) \(7\)
(ii) \(20\,000\) songs
Common pitfalls
Frequently asked questions
What is the index law for multiplying?
Same base: add the indices. \(y^3 \times y^4 = y^7\).
What is the index law for dividing?
Same base: subtract the indices. \(m^8 \div m^3 = m^5\).
What do you do with the numbers in front?
Multiply or divide them normally: \(6k^5 \div 2k^2 = 3k^3\).
Can you use the laws with different bases?
No. Work on each base separately.
What index does a pronumeral on its own have?
\(1\): \(x = x^1\).