Math worksheet for simplifying rational functions through multiplication and division.

Let’s solve each problem one by one. We’ll simplify each rational function step by step, using rules for multiplying and dividing fractions — especially flipping the second fraction when dividing, then canceling common factors.

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Problem 1:
f(x) = (7x² / 12x) ÷ (14x³ / 48y³)

Step 1: Flip the second fraction and multiply:
= (7x² / 12x) × (48y³ / 14x³)

Step 2: Multiply numerators and denominators:
Numerator: 7x² × 48y³ = 336 x² y³
Denominator: 12x × 14x³ = 168 x⁴

So we have: 336 x² y³ / 168 x⁴

Step 3: Simplify coefficients: 336 ÷ 168 = 2
Simplify variables: x² / x⁴ = 1/x²

Final: 2y³ / x²

✔ Check: Yes, correct.

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Problem 2:
f(x) = (24x³ / 50x) · (30 / 8x²)

Step 1: Multiply straight across:
Numerator: 24x³ × 30 = 720 x³
Denominator: 50x × 8x² = 400 x³

So: 720 x³ / 400 x³

Step 2: Cancel x³ top and bottom → leaves 720/400

Step 3: Simplify 720/400 → divide numerator and denominator by 80 → 9/5

Wait — let’s do it more carefully with factoring:

24/50 = 12/25
30/8 = 15/4
So overall: (12/25) × (15/4) × (x³ / x·x²) = (12×15)/(25×4) × x³/x³ = (180/100) × 1 = 9/5

Yes! Final answer: 9/5

✔ Check: Correct.

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Problem 3:
f(x) = (2x²y / y²z) · (y / x)

Step 1: Multiply numerators: 2x²y × y = 2x² y²
Multiply denominators: y²z × x = x y² z

So: 2x² y² / (x y² z)

Step 2: Cancel common terms:
- x² / x = x
- y² / y² = 1

Left with: 2x / z

✔ Check: Correct.

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Problem 4:
f(x) = [(y² - 2y - 15)/4] · [8/(y + 3)]

Step 1: Factor the quadratic: y² - 2y - 15 = (y - 5)(y + 3)

So now: [(y - 5)(y + 3)/4] · [8/(y + 3)]

Step 2: Cancel (y + 3) top and bottom (as long as y ≠ -3):

Left with: (y - 5)/4 × 8/1 = 8(y - 5)/4 = 2(y - 5)

Final: 2y - 10

✔ Check: Correct.

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Problem 5:
f(x) = [(x² - 2x)/6] ÷ [(3x - 6)/x]

Step 1: Flip the second fraction and multiply:
= [(x² - 2x)/6] × [x/(3x - 6)]

Step 2: Factor where possible:
x² - 2x = x(x - 2)
3x - 6 = 3(x - 2)

So: [x(x - 2)/6] × [x / 3(x - 2)]

Step 3: Cancel (x - 2) top and bottom (if x ≠ 2):

Left with: [x/6] × [x/3] = x² / 18

✔ Check: Correct.

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Problem 6:
f(x) = (24x³ / 25y⁵) · (15y² / 8x²)

Step 1: Multiply numerators: 24x³ × 15y² = 360 x³ y²
Multiply denominators: 25y⁵ × 8x² = 200 x² y⁵

So: 360 x³ y² / 200 x² y⁵

Step 2: Simplify coefficients: 360/200 = 9/5 (divide by 40)
Variables: x³/x² = x; y²/y⁵ = 1/y³

Final: (9x)/(5y³)

✔ Check: Correct.

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Problem 7:
f(x) = [(6x - 18)/(4x)] · [x/(2x - 6)]

Step 1: Factor:
6x - 18 = 6(x - 3)
2x - 6 = 2(x - 3)

So: [6(x - 3)/(4x)] · [x / 2(x - 3)]

Step 2: Cancel (x - 3) and x:

Left with: 6/4 × 1/2 = (6×1)/(4×2) = 6/8 = 3/4

✔ Check: Correct.

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Problem 8:
f(x) = [(x² - x - 12)/(4x + 12)] ÷ [(x² - 6x + 8)/6]

Step 1: Flip the second fraction and multiply:
= [(x² - x - 12)/(4x + 12)] × [6/(x² - 6x + 8)]

Step 2: Factor everything:

x² - x - 12 = (x - 4)(x + 3)
4x + 12 = 4(x + 3)
x² - 6x + 8 = (x - 4)(x - 2)

So: [(x - 4)(x + 3) / 4(x + 3)] × [6 / ((x - 4)(x - 2))]

Step 3: Cancel common terms:

Cancel (x + 3): left with (x - 4)/4 × 6/[(x - 4)(x - 2)]
Cancel (x - 4): left with 1/4 × 6/(x - 2) = 6/[4(x - 2)] = 3/[2(x - 2)]

Final: 3 / [2(x - 2)] or 3/(2x - 4)

✔ Check: Correct.

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Final Answers:

1. 2y³ / x²
2. 9/5
3. 2x / z
4. 2y - 10
5. x² / 18
6. 9x / (5y³)
7. 3/4
8. 3 / [2(x - 2)]