Math worksheet for adding and subtracting rational expressions with algebraic fractions.

Let's solve each of the problems on the worksheet step by step. The goal is to add or subtract rational expressions by combining them using common denominators, simplifying where possible.

---

1)

$$
\frac{5d - 3n}{7d^2n^4} + \frac{5d + 5n}{7d^2n^4}
$$

- Same denominator → add numerators:
$$
\frac{(5d - 3n) + (5d + 5n)}{7d^2n^4} = \frac{10d + 2n}{7d^2n^4}
$$

✔ Final Answer:
$$
\boxed{\frac{10d + 2n}{7d^2n^4}}
$$

---

2)

$$
\frac{2k + 8r}{7k^2} - \frac{5k - 5r}{7k^2}
$$

- Same denominator → subtract numerators:
$$
\frac{(2k + 8r) - (5k - 5r)}{7k^2} = \frac{2k + 8r - 5k + 5r}{7k^2} = \frac{-3k + 13r}{7k^2}
$$

✔ Final Answer:
$$
\boxed{\frac{-3k + 13r}{7k^2}}
$$

---

3)

$$
\frac{7h^2 - 8}{2h^3 - 4h^2} - \frac{8h^2 - 4}{2h^3 - 4h^2}
$$

- Same denominator → subtract numerators:
$$
\frac{(7h^2 - 8) - (8h^2 - 4)}{2h^3 - 4h^2} = \frac{7h^2 - 8 - 8h^2 + 4}{2h^3 - 4h^2} = \frac{-h^2 - 4}{2h^3 - 4h^2}
$$

Factor numerator and denominator:

Numerator: $-h^2 - 4 = -(h^2 + 4)$
Denominator: $2h^3 - 4h^2 = 2h^2(h - 2)$

No common factors → cannot simplify further.

✔ Final Answer:
$$
\boxed{\frac{-(h^2 + 4)}{2h^2(h - 2)}}
$$

---

4)

$$
\frac{6c^2 + 9}{7c^3 - 8} - \frac{2c}{7c^3 - 8}
$$

- Same denominator → subtract numerators:
$$
\frac{6c^2 + 9 - 2c}{7c^3 - 8} = \frac{6c^2 - 2c + 9}{7c^3 - 8}
$$

Check if numerator or denominator can be factored:

- Numerator: $6c^2 - 2c + 9$ → no real roots (discriminant: $(-2)^2 - 4(6)(9) = 4 - 216 = -212$), so it doesn’t factor.
- Denominator: $7c^3 - 8$ → not a standard difference of cubes (since $7c^3$ isn't a perfect cube). So leave as is.

✔ Final Answer:
$$
\boxed{\frac{6c^2 - 2c + 9}{7c^3 - 8}}
$$

---

5)

$$
\frac{8s}{7} - \frac{6s + 2}{3s + 6}
$$

First, simplify second term’s denominator:
$$
3s + 6 = 3(s + 2)
$$

So:
$$
\frac{8s}{7} - \frac{6s + 2}{3(s + 2)}
$$

Find LCD: $7 \cdot 3(s + 2) = 21(s + 2)$

Rewrite both fractions:

1st term:
$$
\frac{8s}{7} = \frac{8s \cdot 3(s + 2)}{21(s + 2)} = \frac{24s(s + 2)}{21(s + 2)}
$$

2nd term:
$$
\frac{6s + 2}{3(s + 2)} = \frac{(6s + 2) \cdot 7}{21(s + 2)} = \frac{42s + 14}{21(s + 2)}
$$

Now subtract:
$$
\frac{24s(s + 2) - (42s + 14)}{21(s + 2)} = \frac{24s^2 + 48s - 42s - 14}{21(s + 2)} = \frac{24s^2 + 6s - 14}{21(s + 2)}
$$

Simplify numerator: factor out 2:
$$
= \frac{2(12s^2 + 3s - 7)}{21(s + 2)}
$$

Check if $12s^2 + 3s - 7$ factors:
Discriminant: $9 + 336 = 345$, not a perfect square → no.

So final answer:

✔ Final Answer:
$$
\boxed{\frac{24s^2 + 6s - 14}{21(s + 2)}}
$$

---

6)

$$
\frac{5q}{4q + 3} - \frac{3}{8q + 7}
$$

Different denominators: $4q + 3$ and $8q + 7$

LCD: $(4q + 3)(8q + 7)$

Rewrite:

1st term:
$$
\frac{5q(8q + 7)}{(4q + 3)(8q + 7)}
$$

2nd term:
$$
\frac{3(4q + 3)}{(8q + 7)(4q + 3)}
$$

Subtract:
$$
\frac{5q(8q + 7) - 3(4q + 3)}{(4q + 3)(8q + 7)} = \frac{40q^2 + 35q - 12q - 9}{(4q + 3)(8q + 7)} = \frac{40q^2 + 23q - 9}{(4q + 3)(8q + 7)}
$$

Check if numerator factors:
Try factoring $40q^2 + 23q - 9$

Use AC method: $AC = 40 \cdot (-9) = -360$

Find two numbers that multiply to -360 and add to 23:
Try $45$ and $-8$: $45 \cdot (-8) = -360$, $45 - 8 = 37$ → no
Try $36$ and $-10$: $36 \cdot (-10) = -360$, $36 - 10 = 26$ → no
Try $40$ and $-9$: $40 - 9 = 31$ → no
Try $30$ and $-12$: $30 - 12 = 18$ → no
Try $24$ and $-15$: $24 - 15 = 9$ → no
Try $20$ and $-18$: $20 - 18 = 2$ → no
Try $15$ and $-24$: $15 - 24 = -9$ → no

No obvious factorization → leave as is.

✔ Final Answer:
$$
\boxed{\frac{40q^2 + 23q - 9}{(4q + 3)(8q + 7)}}
$$

---

7)

$$
\frac{7p^3 - 4q^2}{8p^3q^4} - \frac{7p^3 - 3q^2}{8p^3q^4}
$$

Same denominator → subtract numerators:
$$
\frac{(7p^3 - 4q^2) - (7p^3 - 3q^2)}{8p^3q^4} = \frac{7p^3 - 4q^2 - 7p^3 + 3q^2}{8p^3q^4} = \frac{-q^2}{8p^3q^4}
$$

Simplify:
$$
\frac{-q^2}{8p^3q^4} = \frac{-1}{8p^3q^2}
$$

✔ Final Answer:
$$
\boxed{\frac{-1}{8p^3q^2}}
$$

---

8)

$$
\frac{4x^2 + 2s^2}{3x^4} - \frac{6x^2 + 4s^2}{3x^4}
$$

Same denominator → subtract numerators:
$$
\frac{(4x^2 + 2s^2) - (6x^2 + 4s^2)}{3x^4} = \frac{4x^2 + 2s^2 - 6x^2 - 4s^2}{3x^4} = \frac{-2x^2 - 2s^2}{3x^4}
$$

Factor numerator:
$$
= \frac{-2(x^2 + s^2)}{3x^4}
$$

✔ Final Answer:
$$
\boxed{\frac{-2(x^2 + s^2)}{3x^4}}
$$

---

9)

$$
\frac{3b^2 + 2}{6b^3 - 8b^2} - \frac{2b^2 + 8}{6b^3 - 8b^2}
$$

Same denominator → subtract numerators:
$$
\frac{(3b^2 + 2) - (2b^2 + 8)}{6b^3 - 8b^2} = \frac{3b^2 + 2 - 2b^2 - 8}{6b^3 - 8b^2} = \frac{b^2 - 6}{6b^3 - 8b^2}
$$

Factor denominator:
$$
6b^3 - 8b^2 = 2b^2(3b - 4)
$$

Numerator: $b^2 - 6$ → doesn’t factor nicely.

No common factors.

✔ Final Answer:
$$
\boxed{\frac{b^2 - 6}{2b^2(3b - 4)}}
$$

---

10)

$$
\frac{7z^2 - 3}{6z^2 + 4} - \frac{8z}{6z^2 + 4}
$$

Same denominator → subtract numerators:
$$
\frac{7z^2 - 3 - 8z}{6z^2 + 4} = \frac{7z^2 - 8z - 3}{6z^2 + 4}
$$

Factor numerator and denominator:

Numerator: $7z^2 - 8z - 3$

Try factoring:
AC = $7 \cdot (-3) = -21$
Find two numbers: $-11$ and $3$: $-11 + 3 = -8$, $-11 \cdot 3 = -33$ → no
Try $-9$ and $1$: $-9 + 1 = -8$, $-9 \cdot 1 = -9$ → no
Try $-7$ and $3$: $-7 + 3 = -4$ → no
Try $-3$ and $7$: $-3 + 7 = 4$ → no
Try $-21$ and $1$: $-21 + 1 = -20$ → no

Wait: try $-11$ and $3$? No. Try actual factoring:

Try:
$(7z + 1)(z - 3) = 7z^2 - 21z + z - 3 = 7z^2 - 20z - 3$ → no
$(7z - 3)(z + 1) = 7z^2 + 7z - 3z - 3 = 7z^2 + 4z - 3$ → no
$(7z + 3)(z - 1) = 7z^2 - 7z + 3z - 3 = 7z^2 - 4z - 3$ → no
$(7z - 1)(z + 3) = 7z^2 + 21z - z - 3 = 7z^2 + 20z - 3$ → no

Not factorable → leave as is.

Denominator: $6z^2 + 4 = 2(3z^2 + 2)$

No common factors.

✔ Final Answer:
$$
\boxed{\frac{7z^2 - 8z - 3}{6z^2 + 4}}
$$

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## ✔ Summary of Answers:

| Problem | Answer |
|--------|--------|
| 1) | $\frac{10d + 2n}{7d^2n^4}$ |
| 2) | $\frac{-3k + 13r}{7k^2}$ |
| 3) | $\frac{-(h^2 + 4)}{2h^2(h - 2)}$ |
| 4) | $\frac{6c^2 - 2c + 9}{7c^3 - 8}$ |
| 5) | $\frac{24s^2 + 6s - 14}{21(s + 2)}$ |
| 6) | $\frac{40q^2 + 23q - 9}{(4q + 3)(8q + 7)}$ |
| 7) | $\frac{-1}{8p^3q^2}$ |
| 8) | $\frac{-2(x^2 + s^2)}{3x^4}$ |
| 9) | $\frac{b^2 - 6}{2b^2(3b - 4)}$ |
| 10) | $\frac{7z^2 - 8z - 3}{6z^2 + 4}$ |

Let me know if you'd like these simplified further or checked with a calculator!