Dividing Terms
Every question with a fully worked solution.
- Dividing Terms - Video - Dividing Terms - Example 1 Watch
Theory
To divide terms, divide (or simplify) the numbers, then cancel the pronumerals that are on both the top and the bottom. With powers of the same pronumeral, subtract the indices: \(9x^3 \div 3x = 3x^{3-1} = 3x^2\). This Year 8 maths page shows how.
Write the division as a fraction, first term on top: \(20cd \div 5c = \dfrac{20cd}{5c}\).
Expand the top and bottom so every factor shows. Divide the numbers by their HCF. Cancel each pronumeral that is on both the top and the bottom, because a pronumeral divided by itself is \(1\). A pronumeral only on the top stays on top; one only on the bottom stays on the bottom.
Powers. The same pronumeral to different powers: subtract the indices, \(x^3 \div x^1 = x^{3-1} = x^2\). If the bigger power is on the bottom, what is left stays on the bottom: \(\dfrac{k}{k^3} = \dfrac{1}{k^2}\).
Signs. Same signs give a positive answer; different signs a negative one. A minus sign in front of a fraction belongs to the whole fraction.
Order. The fraction bar groups: work out a product on top first. \(\times\) and \(\div\) are done left to right, before \(+\) and \(-\): \(12 - 9m \div 3 = 12 - 3m\).
What cancels:
| Fraction | Expanded | Cancels to |
|---|---|---|
| \(\dfrac{k}{k}\) | \(\dfrac{k}{k}\) | \(1\) |
| \(\dfrac{k^3}{k}\) | \(\dfrac{k \times k \times k}{k}\) | \(k^2\) |
| \(\dfrac{k}{k^3}\) | \(\dfrac{k}{k \times k \times k}\) | \(\dfrac{1}{k^2}\) |
How to divide terms
- Write it as a fraction and decide the sign.
- Divide the numbers by their HCF.
- Cancel pronumerals (subtract the indices of a repeated pronumeral).
- Check by substituting numbers.
| \(9x^3 \div 3x\) | \(=\) | \((9 \div 3) \times x^3 \div x^1\) |
| \(\) | \(=\) | \(3x^{3-1} = 3x^2\) |
\(3x^2\)
| \(8x^2y \div 4x\) | \(=\) | \((8 \div 4) \times (x^2 \div x) \times y\) |
| \(\) | \(=\) | \(2xy\) |
\(2xy\) (the \(y\) is only on the top)
| \(\dfrac{15yx^2}{3xy}\) | \(=\) | \(5 \times \dfrac{x^2y}{xy}\) |
| \(\) | \(=\) | \(5x\) |
\(5x\): one \(x\) and the \(y\) cancel.
(i) \(\dfrac{-28k^2}{7k}\)
(ii) \(\dfrac{-30w}{-5w}\)
| \(\dfrac{-28k^2}{7k}\) | \(=\) | \(-4k\) |
| \(\dfrac{-30w}{-5w}\) | \(=\) | \(6\) |
(i) \(-4k\). Check with \(k = 3\): \(-252 \div 21 = -12\).
(ii) \(6\) (every pronumeral cancels)
| \(\dfrac{18e^2f}{4ef^2}\) | \(=\) | \(\dfrac{18 \times e \times e \times f}{4 \times e \times f \times f}\) |
| \(\) | \(=\) | \(\dfrac{9e}{2f}\) |
\(\dfrac{9e}{2f}\). Check with \(e = 2\), \(f = 3\): \(216 \div 72 = 3\) and \(\dfrac{18}{6} = 3\).
(i) \(18n \div (-6) \times 2n\)
(ii) \(20p^2 - \dfrac{15p^3}{5p}\)
| \(18n \div (-6)\) | \(=\) | \(-3n\) |
| \(-3n \times 2n\) | \(=\) | \(-6n^2\) |
| \(20p^2 - 3p^2\) | \(=\) | \(17p^2\) |
(i) \(-6n^2\) (left to right)
(ii) \(\dfrac{15p^3}{5p} = 3p^2\), so \(17p^2\)
Common pitfalls
Frequently asked questions
How do you divide algebraic terms?
Write it as a fraction, divide the numbers, then cancel pronumerals: \(12ab \div 4a = 3b\).
What is x cubed divided by x?
\(x^2\): subtract the indices, \(3 - 1 = 2\).
What happens when the same term is on the top and the bottom?
It cancels to \(1\): \(\dfrac{5y}{5y} = 1\).
Can the answer be a fraction?
Yes: \(14m \div 4n = \dfrac{7m}{2n}\).
What is a negative divided by a negative?
Positive: \(\dfrac{-12t}{-4} = 3t\).