Simplify complex rational expressions worksheet from Math Monks.

Let’s solve each problem one by one.

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Problem 1:

We have:
  6 / (x - 4)
 ──────────────
 3 / (x² - 16)

Step 1: Notice that x² - 16 is a difference of squares → (x - 4)(x + 4)

So rewrite the denominator:
  6 / (x - 4)
 ─────────────────
 3 / [(x - 4)(x + 4)]

Step 2: Dividing fractions = multiply by reciprocal:

= [6 / (x - 4)] × [(x - 4)(x + 4) / 3]

Step 3: Cancel common terms:

- (x - 4) cancels top and bottom
- 6 ÷ 3 = 2

Left with: 2(x + 4)

✔ Final for #1: 2(x + 4)

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Problem 2:

We have:
  2x / (27y²)
 ──────────────
 6x² / 9

Step 1: Rewrite as multiplication by reciprocal:

= [2x / (27y²)] × [9 / (6x²)]

Step 2: Multiply numerators and denominators:

Numerator: 2x × 9 = 18x
Denominator: 27y² × 6x² = 162 x² y²

→ 18x / (162 x² y²)

Step 3: Simplify coefficients: 18/162 = 1/9

Simplify variables: x / x² = 1/x

So: 1/(9 x y²)

✔ Final for #2: 1/(9xy²)

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Problem 3:

We have:
  3 / (x² - 16)
 ───────────────
 1 / (x + 4)

Again, x² - 16 = (x - 4)(x + 4)

Rewrite:

= [3 / ((x - 4)(x + 4))] × [(x + 4)/1]

Cancel (x + 4):

Left with: 3 / (x - 4)

✔ Final for #3: 3/(x - 4)

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Problem 4:

We have:
  1 - 1/x²
 ────────────
 1 - 1/x

Step 1: Combine terms in numerator and denominator.

Numerator: 1 - 1/x² = (x² - 1)/x²
Denominator: 1 - 1/x = (x - 1)/x

So now we have:

[(x² - 1)/x²] ÷ [(x - 1)/x]

Step 2: Multiply by reciprocal:

= [(x² - 1)/x²] × [x/(x - 1)]

Step 3: Factor x² - 1 = (x - 1)(x + 1)

So:

= [(x - 1)(x + 1)/x²] × [x/(x - 1)]

Step 4: Cancel (x - 1), and cancel one x from x² and x:

Left with: (x + 1)/x

✔ Final for #4: (x + 1)/x

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Problem 5:

We have:
  x/y² + 1/y
 ─────────────
 y/x² + 1/x

Step 1: Get common denominators in numerator and denominator.

Numerator: x/y² + 1/y = x/y² + y/y² = (x + y)/y²
Denominator: y/x² + 1/x = y/x² + x/x² = (y + x)/x²

Note: x + y = y + x → same thing.

So expression becomes:

[(x + y)/y²] ÷ [(x + y)/x²]

Step 2: Multiply by reciprocal:

= [(x + y)/y²] × [x²/(x + y)]

Step 3: Cancel (x + y)

Left with: x² / y²

✔ Final for #5: x²/y²

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Problem 6:

We have:
  1/x + 1/y
 ─────────────
 x/y - y/x

Step 1: Common denominators.

Numerator: 1/x + 1/y = (y + x)/(xy)
Denominator: x/y - y/x = (x² - y²)/(xy) ← because LCD is xy: (x·x - y·y)/xy

Wait — let’s do it carefully:

x/y = x²/(xy)? No! To get denominator xy:

x/y = (x * x) / (y * x) ? Wait no — better:

To write x/y with denominator xy: multiply numerator and denominator by x → x²/(xy)

Similarly, y/x = y²/(xy)

So: x/y - y/x = (x² - y²)/xy

Yes.

So overall:

[(x + y)/xy] ÷ [(x² - y²)/xy]

Step 2: Multiply by reciprocal:

= [(x + y)/xy] × [xy/(x² - y²)]

Step 3: Cancel xy

Left with: (x + y)/(x² - y²)

But x² - y² = (x - y)(x + y)

So: (x + y)/[(x - y)(x + y)] = 1/(x - y)

✔ Final for #6: 1/(x - y)

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Final Answers:

1. 2(x + 4)
2. 1/(9xy²)
3. 3/(x - 4)
4. (x + 1)/x
5. x²/y²
6. 1/(x - y)

──────────────────────────────────────

Final Answer:
1. 2(x + 4)
2. \frac{1}{9xy^2}
3. \frac{3}{x - 4}
4. \frac{x + 1}{x}
5. \frac{x^2}{y^2}
6. \frac{1}{x - y}