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 ÷ 3x² | 12 ÷ 3 = 4 | 5 − 2 = 3 | 4x³ |
| 15a³b² ÷ 5ab | 15 ÷ 5 = 3 | a: 3−1=2, b: 2−1=1 | 3a²b |
| 8x² ÷ 2x² | 8 ÷ 2 = 4 | 2 − 2 = 0 | 4 |
| 6x ÷ 12x² | 6 ÷ 12 = ½ | 1 − 2 = −1 | 1/(2x) |
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.
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 just means more copies below than above. Write 1 over b² rather than b to the power −2.
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