Dividing Algebraic Terms
Division is multiplication's undo button. Learn to divide coefficients, cancel variables, and simplify algebraic fractions.
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Simplify $\frac{8x}{2x}$. Explain what happens to the numbers and what happens to the letters.
To divide algebraic terms: divide the coefficients, and cancel or subtract powers for the variables. It's the opposite of multiplication.
$\frac{8x}{2x}$: coefficients $8 \div 2 = 4$, variables $\frac{x}{x} = 1$. So $4 \times 1 =$ $4$. The $x$ cancels out!
Know
- Divide coefficients separately
- Cancel matching variables
- Subtract powers for same base
Understand
- Why $\frac{x}{x} = 1$ not $0$
- That division undoes multiplication
- How to handle leftover variables
Can Do
- Divide simple algebraic terms
- Simplify algebraic fractions
- Handle negative coefficients
Wrong: $\frac{6x}{2} = 3x$ is correct, but $\frac{6x}{2x} = 3x$ is NOT.
Right: $\frac{6x}{2x} = 3$. The $x$ cancels, leaving just 3.
Wrong: $\frac{x^5}{x^2} = x^{5 \div 2} = x^{2.5}$
Right: $\frac{x^5}{x^2} = x^{5-2} = x^3$. Subtract, don't divide!
When dividing same variables, subtract the powers. $\frac{x^m}{x^n} = x^{m-n}$.
Divide coefficients. Cancel each variable separately.
| Expression | Coefficients | Variables | Result |
|---|---|---|---|
| $\frac{15x^3y}{3xy}$ | $15 \div 3 = 5$ | $x^{3-1} = x^2$, $y^{1-1} = 1$ | $5x^2$ |
| $\frac{-8a^2b^3}{2ab^2}$ | $-8 \div 2 = -4$ | $a^{2-1} = a$, $b^{3-2} = b$ | $-4ab$ |
| $\frac{6x^2y}{3x^2}$ | $6 \div 3 = 2$ | $x^{2-2} = 1$, $y$ stays | $2y$ |
If a variable has the same power on top and bottom, it becomes 1 and disappears.
| Division | Working | Answer |
|---|---|---|
| $\frac{12x}{3}$ | $12 \div 3 = 4$, $x$ stays | $4x$ |
| $\frac{9x^2}{3x}$ | $9 \div 3 = 3$, $x^{2-1}$ | $3x$ |
| $\frac{-15a^3}{5a}$ | $-15 \div 5 = -3$, $a^{3-1}$ | $-3a^2$ |
| $\frac{8x^2y}{2xy}$ | $8 \div 2 = 4$, $x^{2-1}$, $y^{1-1}=1$ | $4x$ |
Watch Me Solve It · Worked example
- 1Divide the coefficients$-20 \div 4 = -5$Negative divided by positive gives negative. $20 \div 4 = 5$, so answer is $-5$.
- 2Divide the $x$ variables$\frac{x^3}{x^2} = x^{3-2} = x^1 = x$Subtract powers: $3 - 2 = 1$. $x^1$ is just $x$.
- 3Divide the $y$ variables$\frac{y^2}{y} = y^{2-1} = y^1 = y$$y = y^1$. Subtract: $2 - 1 = 1$. Just $y$ remains.
- 4Combine all parts$-5 \times x \times y = -5xy$Coefficient $-5$, $x$-part $x$, $y$-part $y$. Write together: $-5xy$.
How are you completing this lesson?
Brain Trainer · 4 problems
1 Simplify $\frac{12x}{3}$.
$12 \div 3 = 4$, $x$ stays. $4x$$4x$2 Simplify $\frac{15x^3}{5x}$.
$15 \div 5 = 3$, $x^{3-1} = x^2$. $3x^2$$3x^2$3 Simplify $\frac{8x^2y}{2xy}$.
$8 \div 2 = 4$, $x^{2-1} = x$, $y^{1-1} = 1$. $4x$$4x$4 Simplify $\frac{-18a^4b^2}{3a^2b}$.
$-18 \div 3 = -6$, $a^{4-2} = a^2$, $b^{2-1} = b$. $-6a^2b$$-6a^2b$
Show Your Working · 3 questions
Q6. Simplify $\frac{18x^2}{6x}$.
Q7. Simplify $\frac{-20a^3b^2}{4ab}$.
Q8. A triangle has area $\frac{1}{2} \times \text{base} \times \text{height}$. If the area is $15x^2$, the base is $5x$, find the height.
Stretch. If $\frac{30x^3}{kx} = 6x^2$, find the value of $k$.
Cancel factors, not letters at random
Cancel factors, not letters at random. For positive $a$ and $b$, a student simplifies $24a^2b\div6ab$ to $4ab$.
Correct the simplification and explain what happens to each factor. Check the original quotient and your answer when $a=3$ and $b=2$.
Show your working first, then open the worked solution to check your reasoning.
Check the worked solution
Write the quotient as $\frac{24aab}{6ab}$. The numerical factor becomes 4, one $a$ cancels and the $b$ factors cancel, leaving $4a$. The positive-value assumption ensures the denominator is nonzero. With $a=3$ and $b=2$, the quotient is $432\div36=12$, and $4a=12$. The student’s $4ab$ would give 24, so an extra $b$ was left behind.
Key Concept
Review the main ideas from this lesson.
Formulas
Key formulas and rules.
Watch Out
Common mistakes to avoid.
Check
Always verify your answers.
Practice
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Next
Build on these skills.
Interactive: Algebra Machine
Substitute numbers into algebraic expressions and see them evaluate step by step.
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