Rational Expressions • Topic 3 of 6

Division

To divide rational expressions, multiply by the reciprocal of the second fraction (flip and multiply), then factor and cancel. So (3/x) ÷ (9/x²) = (3/x)(x²/9) = 3x²/(9x) = x/3. Keep, change, flip: keep the first fraction, change ÷ to ×, flip the divisor. After flipping, cancel any common factors before multiplying. Watch restrictions: values that make any denominator — including the original divisor's numerator — zero are excluded. Division questions on the SAT test whether you correctly take the reciprocal and then simplify, so the flip step is where care pays off.

✅ Solved examples

1. Simplify (3/x) ÷ (9/x²).
(3/x)(x²/9) = 3x²/(9x) = x/3.
2. Simplify (x/2) ÷ (x/8).
(x/2)(8/x) = 8x/(2x) = 4.
3. Simplify (6/x) ÷ (2/x).
(6/x)(x/2) = 6x/(2x) = 3.
4. Simplify (4/x²) ÷ (2/x).
(4/x²)(x/2) = 4x/(2x²) = 2/x.

✏️ Practice — try these, take hints as needed

1. Simplify (x/3) ÷ (x/12).
Multiply by the reciprocal: (x/3)(12/x).
12x/(3x).
—
4.
2. Simplify (8/x) ÷ (4/x).
(8/x)(x/4).
Cancel x.
—
2.
3. Simplify (x²/2) ÷ (x/6).
(x²/2)(6/x).
6x²/(2x).
—
3x.
4. Simplify (10/x) ÷ (5/x²).
(10/x)(x²/5).
10x²/(5x).
—
2x.
5. Simplify (1/x) ÷ (1/x²).
(1/x)(x²/1).
x²/x.
—
x.

📝 Topic test — 8 questions

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