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📐 Mathematics - Class 7
Lesson 22 · Algebraic Expressions

Dividing Algebraic Expressions

MathematicsSubject
13 minEstimated read
Multiplication run backwards

The coefficients divide and the powers subtract. You already know the rules; today you use them in reverse.

Why the powers subtract

x⁵ ÷ x² is five x's over two x's. Two cancel, leaving three, so the answer is x³.

Division Coefficients Powers Answer
12x⁵ ÷ 3x²12 ÷ 3 = 45 − 2 = 34x³
15a³b² ÷ 5ab15 ÷ 5 = 3a: 3−1=2, b: 2−1=13a²b
8x² ÷ 2x²8 ÷ 2 = 42 − 2 = 04
6x ÷ 12x²6 ÷ 12 = ½1 − 2 = −11/(2x)
Anything to the power zero is 1

8x² ÷ 2x² = 4. The x² cancels completely, leaving 1 behind and not 0.

Dividing an expression by a single term

Every term on top must be divided. (6x + 9) ÷ 3 is 2x + 3, because both terms get divided.

You cannot cancel a term, only a factor

Cancelling only works on something that multiplies the WHOLE top. Test with x = 1: the original gives 5, and 2x + 3 gives 5 while 6x + 3 gives 9.

Simplifying a fraction of terms

Deal with the numbers and each letter separately. 20a⁴b³ ÷ 8a²b⁵ gives 5a² over 2b².

A negative power means the letter belongs underneath

A negative power just means more copies below than above. Write 1 over b² rather than b to the power −2.

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Worked Example 1

Simplify 15a³b² ÷ 5ab.

Numbers first15 ÷ 5 = 3
Now each letter in turna: 3 − 1 = 2. b: 2 − 1 = 1
Put it together3a²b
Check by multiplying back3a²b × 5ab returns 15a³b². This check always works.
Worked Example 2

Simplify (12x³ − 8x² + 4x) ÷ 4x.

Divide each term separatelyThree terms on top means three terms in the answer
First term12x³ ÷ 4x = 3x²
Second and third terms−8x² ÷ 4x = −2x, and 4x ÷ 4x = 1
Answer and check3x² − 2x + 1. With x = 1 both give 2.
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Activity 22.3

The cancelling strips

What you needSmall paper cards each with a single x on it, and a sheet with a horizontal line across the middle.
MethodLay five cards above the line and two below, then remove one from each side at a time.
Then try the awkward casesTry equal numbers on both sides, and then more cards below than above.
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1

Division undoes multiplication: the coefficients DIVIDE and the powers SUBTRACT.

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1Why do the powers subtract when you divide?
2Why is 8x² divided by 2x² equal to 4 and not 0?
3Why must every term on top be divided, and what goes wrong if you cancel?
4What does a negative power mean when you simplify a fraction?
5How can you check an algebraic division without doing it again?

Question 1 of 10

1When dividing algebraic terms, the powers
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Slide 1 of 4
Unit 22 · Division

Dividing Expressions

Mathematics · Grade 7

What You Will Be Able To Do

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Divide one term by another, handling the powers correctly

0️⃣

Say why a power of zero leaves 1 behind and not 0

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Divide an expression of several terms by a single term

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Know what may be cancelled in a fraction, and what may not

“

Multiplying meant the coefficients multiply and the powers add, so dividing must undo both. The coefficients divide and the powers subtract.

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Presenter notes: Open by asking the class to predict today lesson before it starts. Remind them that multiplying meant the coefficients multiply and the powers add, then ask what division ought to do. Somebody will say the coefficients divide and the powers subtract, and they will be exactly right. Tell them there is nothing new to learn today, only the same two rules used in reverse.