301 Moved Permanently - Free Printable

Let's solve each problem step by step. The goal is to simplify each expression by combining like terms — terms that have the same variables raised to the same powers.

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1)

$$
(x^3y^3 - 5x^3y) + (-7x^3y + 6x^3y^3) - (5x^3y^2 + 7x^3y + 8x^3y^3)
$$

Step 1: Remove parentheses and distribute the negative sign:

$$
x^3y^3 - 5x^3y - 7x^3y + 6x^3y^3 - 5x^3y^2 - 7x^3y - 8x^3y^3
$$

Step 2: Combine like terms:

- $x^3y^3$: $1 + 6 - 8 = -1$ → $-x^3y^3$
- $x^3y$: $-5 -7 -7 = -19$ → $-19x^3y$
- $x^3y^2$: $-5x^3y^2$

✔ Answer:
$$
-x^3y^3 - 19x^3y - 5x^3y^2
$$

---

2)

$$
(-5x^4y + 8x^4y^4) + (-3x^4y^4 - 7x^4y) - (-4xy + 6x^4y^4 - x^4y^2)
$$

Step 1: Distribute the negative sign:

$$
-5x^4y + 8x^4y^4 - 3x^4y^4 - 7x^4y + 4xy - 6x^4y^4 + x^4y^2
$$

Step 2: Combine like terms:

- $x^4y$: $-5 -7 = -12$ → $-12x^4y$
- $x^4y^4$: $8 - 3 - 6 = -1$ → $-x^4y^4$
- $x^4y^2$: $+x^4y^2$
- $xy$: $+4xy$

✔ Answer:
$$
-12x^4y - x^4y^4 + x^4y^2 + 4xy
$$

---

3)

$$
(-8u^3v + 6u^4v^3) - (-8u^3v - 2uv) + (4uv - 2u^3v - 2u^4v)
$$

Step 1: Distribute negatives:

$$
-8u^3v + 6u^4v^3 + 8u^3v + 2uv + 4uv - 2u^3v - 2u^4v
$$

Step 2: Combine like terms:

- $u^3v$: $-8 + 8 - 2 = -2$ → $-2u^3v$
- $u^4v^3$: $+6u^4v^3$
- $u^4v$: $-2u^4v$
- $uv$: $2 + 4 = 6$ → $6uv$

✔ Answer:
$$
-2u^3v + 6u^4v^3 - 2u^4v + 6uv
$$

---

4)

$$
(8x^2 + 5x^4y^3) + (-x^2 + 4y^3) - (6x^4y^3 - 2y^3 + 4x^2)
$$

Step 1: Distribute negative:

$$
8x^2 + 5x^4y^3 - x^2 + 4y^3 - 6x^4y^3 + 2y^3 - 4x^2
$$

Step 2: Combine:

- $x^2$: $8 -1 -4 = 3$ → $3x^2$
- $x^4y^3$: $5 -6 = -1$ → $-x^4y^3$
- $y^3$: $4 + 2 = 6$ → $6y^3$

✔ Answer:
$$
3x^2 - x^4y^3 + 6y^3
$$

---

5)

$$
(-7ab^2 - 6b) + (6ab^2 - a^3b^2) + (4ab^2 + 7a^4b^3 + 4a^3b^2)
$$

Step 1: Combine all:

$$
-7ab^2 - 6b + 6ab^2 - a^3b^2 + 4ab^2 + 7a^4b^3 + 4a^3b^2
$$

Step 2: Combine:

- $ab^2$: $-7 + 6 + 4 = 3$ → $3ab^2$
- $b$: $-6b$
- $a^3b^2$: $-1 + 4 = 3$ → $3a^3b^2$
- $a^4b^3$: $+7a^4b^3$

✔ Answer:
$$
3ab^2 - 6b + 3a^3b^2 + 7a^4b^3
$$

---

6)

$$
(-7x^3y^3 - 6x^2) + (-8x^3y^3 + 4x^2) + (4x^3y^3 - 3x^2 - 6x^3y^2)
$$

Step 1: Combine:

$$
-7x^3y^3 - 6x^2 - 8x^3y^3 + 4x^2 + 4x^3y^3 - 3x^2 - 6x^3y^2
$$

Step 2: Combine:

- $x^3y^3$: $-7 -8 + 4 = -11$ → $-11x^3y^3$
- $x^2$: $-6 + 4 - 3 = -5$ → $-5x^2$
- $x^3y^2$: $-6x^3y^2$

✔ Answer:
$$
-11x^3y^3 - 5x^2 - 6x^3y^2
$$

---

7)

$$
(2a^3b^2 - 5a^2b) + (4a^3b^2 - 4b^3) + (-2a^3b^2 - 3a^2b - 5b^3)
$$

Step 1: Combine:

$$
2a^3b^2 - 5a^2b + 4a^3b^2 - 4b^3 - 2a^3b^2 - 3a^2b - 5b^3
$$

Step 2: Combine:

- $a^3b^2$: $2 + 4 - 2 = 4$ → $4a^3b^2$
- $a^2b$: $-5 -3 = -8$ → $-8a^2b$
- $b^3$: $-4 -5 = -9$ → $-9b^3$

✔ Answer:
$$
4a^3b^2 - 8a^2b - 9b^3
$$

---

8)

$$
(-2x^2y^3 + 8x^3) + (5x^2y^2 + 5x^2y^3) + (7x^4y^3 - 8x^2y^3 - 5x^3)
$$

Step 1: Combine:

$$
-2x^2y^3 + 8x^3 + 5x^2y^2 + 5x^2y^3 + 7x^4y^3 - 8x^2y^3 - 5x^3
$$

Step 2: Combine:

- $x^2y^3$: $-2 + 5 - 8 = -5$ → $-5x^2y^3$
- $x^3$: $8 - 5 = 3$ → $3x^3$
- $x^2y^2$: $+5x^2y^2$
- $x^4y^3$: $+7x^4y^3$

✔ Answer:
$$
7x^4y^3 - 5x^2y^3 + 5x^2y^2 + 3x^3
$$

---

9)

$$
(7 + 3m^3n^2) - (-3m^2n^4 - 4) - (-3m^2n^4 + m^3n^2 - 4)
$$

Step 1: Distribute negatives:

$$
7 + 3m^3n^2 + 3m^2n^4 + 4 + 3m^2n^4 - m^3n^2 + 4
$$

Step 2: Combine:

- Constants: $7 + 4 + 4 = 15$
- $m^3n^2$: $3 - 1 = 2$ → $2m^3n^2$
- $m^2n^4$: $3 + 3 = 6$ → $6m^2n^4$

✔ Answer:
$$
15 + 2m^3n^2 + 6m^2n^4
$$

---

10)

$$
(2y^2 + 3x^2y^4) + (-3x^2y^4 - 3y^2) + (-6x^3y^2 + 8x^2 - 2y^2)
$$

Step 1: Combine:

$$
2y^2 + 3x^2y^4 - 3x^2y^4 - 3y^2 - 6x^3y^2 + 8x^2 - 2y^2
$$

Step 2: Combine:

- $y^2$: $2 - 3 - 2 = -3$ → $-3y^2$
- $x^2y^4$: $3 - 3 = 0$
- $x^3y^2$: $-6x^3y^2$
- $x^2$: $+8x^2$

✔ Answer:
$$
-3y^2 - 6x^3y^2 + 8x^2
$$

---

11)

$$
(-2x^2y^4 + x^4y^2 + x^3) - (2x^3 + 5x^4y^2) - (7x^2y^4 - 4x^4)
$$

Step 1: Distribute negatives:

$$
-2x^2y^4 + x^4y^2 + x^3 - 2x^3 - 5x^4y^2 - 7x^2y^4 + 4x^4
$$

Step 2: Combine:

- $x^2y^4$: $-2 -7 = -9$ → $-9x^2y^4$
- $x^4y^2$: $1 - 5 = -4$ → $-4x^4y^2$
- $x^3$: $1 - 2 = -1$ → $-x^3$
- $x^4$: $+4x^4$

✔ Answer:
$$
-9x^2y^4 - 4x^4y^2 - x^3 + 4x^4
$$

---

12)

$$
(-2uv^4 + 3uv^3 - uv^2) - (-uv^2 - 4uv^4) - (-4uv^4 - 2uv^3)
$$

Step 1: Distribute negatives:

$$
-2uv^4 + 3uv^3 - uv^2 + uv^2 + 4uv^4 + 4uv^4 + 2uv^3
$$

Step 2: Combine:

- $uv^4$: $-2 + 4 + 4 = 6$ → $6uv^4$
- $uv^3$: $3 + 2 = 5$ → $5uv^3$
- $uv^2$: $-1 + 1 = 0$

✔ Answer:
$$
6uv^4 + 5uv^3
$$

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13)

$$
(-2x^4 + 5x^2y^3 + 5x) + (5xy^2 - 8x^2y^3) + (2x^2y^3 + 8x)
$$

Step 1: Combine:

$$
-2x^4 + 5x^2y^3 + 5x + 5xy^2 - 8x^2y^3 + 2x^2y^3 + 8x
$$

Step 2: Combine:

- $x^4$: $-2x^4$
- $x^2y^3$: $5 - 8 + 2 = -1$ → $-x^2y^3$
- $x$: $5 + 8 = 13$ → $13x$
- $xy^2$: $+5xy^2$

✔ Answer:
$$
-2x^4 - x^2y^3 + 13x + 5xy^2
$$

---

14)

$$
(-8ab^4 + 6a^4b^4 + 5a^2b^2) - (5a^4b^4 - 5ab^4) - (-6a^4b^4 - 3a^2b^2)
$$

Step 1: Distribute negatives:

$$
-8ab^4 + 6a^4b^4 + 5a^2b^2 - 5a^4b^4 + 5ab^4 + 6a^4b^4 + 3a^2b^2
$$

Step 2: Combine:

- $ab^4$: $-8 + 5 = -3$ → $-3ab^4$
- $a^4b^4$: $6 - 5 + 6 = 7$ → $7a^4b^4$
- $a^2b^2$: $5 + 3 = 8$ → $8a^2b^2$

✔ Answer:
$$
-3ab^4 + 7a^4b^4 + 8a^2b^2
$$

---

15)

$$
(-3x^2y - xy^4 - 4y^4) + (-8y^4 + 5xy^4) - (-4y^4 + 5xy^4)
$$

Step 1: Distribute the last negative:

$$
-3x^2y - xy^4 - 4y^4 - 8y^4 + 5xy^4 + 4y^4 - 5xy^4
$$

Step 2: Combine:

- $x^2y$: $-3x^2y$
- $xy^4$: $-1 + 5 - 5 = -1$ → $-xy^4$
- $y^4$: $-4 -8 + 4 = -8$ → $-8y^4$

✔ Answer:
$$
-3x^2y - xy^4 - 8y^4
$$

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✔ Final Answers:

1) $-x^3y^3 - 19x^3y - 5x^3y^2$
2) $-12x^4y - x^4y^4 + x^4y^2 + 4xy$
3) $-2u^3v + 6u^4v^3 - 2u^4v + 6uv$
4) $3x^2 - x^4y^3 + 6y^3$
5) $3ab^2 - 6b + 3a^3b^2 + 7a^4b^3$
6) $-11x^3y^3 - 5x^2 - 6x^3y^2$
7) $4a^3b^2 - 8a^2b - 9b^3$
8) $7x^4y^3 - 5x^2y^3 + 5x^2y^2 + 3x^3$
9) $15 + 2m^3n^2 + 6m^2n^4$
10) $-3y^2 - 6x^3y^2 + 8x^2$
11) $-9x^2y^4 - 4x^4y^2 - x^3 + 4x^4$
12) $6uv^4 + 5uv^3$
13) $-2x^4 - x^2y^3 + 13x + 5xy^2$
14) $-3ab^4 + 7a^4b^4 + 8a^2b^2$
15) $-3x^2y - xy^4 - 8y^4$

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