Pre-Algebra Worksheets | Monomials and Polynomials Worksheets - Free Printable

Let's solve each problem step by step. The goal is to multiply and simplify monomials using the rules of exponents:

Rules Used:

1. Power of a product: $(ab)^n = a^n b^n$
2. Power of a power: $(a^m)^n = a^{m \cdot n}$
3. Product of powers: $a^m \cdot a^n = a^{m+n}$
4. Distributive property over multiplication: $a(bc) = (a \cdot b)c$

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We’ll go through each problem one at a time.

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1) $(2y^5z^2) \cdot (y^5z^2)$

Multiply coefficients and add exponents for like bases:
- Coefficients: $2 \cdot 1 = 2$
- $y^5 \cdot y^5 = y^{5+5} = y^{10}$
- $z^2 \cdot z^2 = z^{2+2} = z^4$

✔ Answer: $2y^{10}z^4$

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2) $(2z^3y)^3 \cdot (z^3)^3$

First, expand $(2z^3y)^3$:
- $= 2^3 \cdot (z^3)^3 \cdot y^3 = 8 \cdot z^9 \cdot y^3 = 8y^3z^9$

Now multiply by $(z^3)^3 = z^9$

So: $8y^3z^9 \cdot z^9 = 8y^3z^{9+9} = 8y^3z^{18}$

✔ Answer: $8y^3z^{18}$

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3) $(2y^3z^3)^3 \cdot (y^3)^3$

First, $(2y^3z^3)^3 = 2^3 \cdot (y^3)^3 \cdot (z^3)^3 = 8 \cdot y^9 \cdot z^9 = 8y^9z^9$

Then $(y^3)^3 = y^9$

Now multiply: $8y^9z^9 \cdot y^9 = 8y^{9+9}z^9 = 8y^{18}z^9$

✔ Answer: $8y^{18}z^9$

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4) $(y^27x^4)^2$

This is $(y^2 \cdot 7x^4)^2 = (7x^4y^2)^2$

Apply power to each part:
- $7^2 = 49$
- $(x^4)^2 = x^8$
- $(y^2)^2 = y^4$

✔ Answer: $49x^8y^4$

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5) $(x^23z^5)^2$

Same as $(3x^2z^5)^2$

- $3^2 = 9$
- $(x^2)^2 = x^4$
- $(z^5)^2 = z^{10}$

✔ Answer: $9x^4z^{10}$

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6) $(3yz^3)^2 \cdot (y^2)^3$

First: $(3yz^3)^2 = 3^2 \cdot y^2 \cdot (z^3)^2 = 9y^2z^6$

Then: $(y^2)^3 = y^6$

Now multiply: $9y^2z^6 \cdot y^6 = 9y^{2+6}z^6 = 9y^8z^6$

✔ Answer: $9y^8z^6$

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7) $4(4xy^5)^3$

First: $(4xy^5)^3 = 4^3 \cdot x^3 \cdot (y^5)^3 = 64x^3y^{15}$

Now multiply by 4: $4 \cdot 64x^3y^{15} = 256x^3y^{15}$

✔ Answer: $256x^3y^{15}$

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8) $4(9x^3y^5)^2$

First: $(9x^3y^5)^2 = 9^2 \cdot (x^3)^2 \cdot (y^5)^2 = 81x^6y^{10}$

Now multiply by 4: $4 \cdot 81x^6y^{10} = 324x^6y^{10}$

✔ Answer: $324x^6y^{10}$

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9) $4(8xz^2)^3$

First: $(8xz^2)^3 = 8^3 \cdot x^3 \cdot (z^2)^3 = 512x^3z^6$

Now multiply by 4: $4 \cdot 512x^3z^6 = 2048x^3z^6$

✔ Answer: $2048x^3z^6$

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10) $(8z^4x^5)^3 \cdot (z^4x^5)^3$

Note: both are same base, so we can write:

$$
(8z^4x^5)^3 \cdot (z^4x^5)^3 = [8(z^4x^5)^3] \cdot [(z^4x^5)^3] = 8 \cdot (z^4x^5)^3 \cdot (z^4x^5)^3
$$

But better: factor as:

$$
= 8 \cdot (z^4x^5)^3 \cdot (z^4x^5)^3 = 8 \cdot (z^4x^5)^6
$$

Now compute $(z^4x^5)^6 = z^{24}x^{30}$

So total: $8z^{24}x^{30}$

Alternatively, compute each separately:

- $(8z^4x^5)^3 = 8^3 \cdot z^{12} \cdot x^{15} = 512z^{12}x^{15}$
- $(z^4x^5)^3 = z^{12}x^{15}$
- Multiply: $512z^{12}x^{15} \cdot z^{12}x^{15} = 512z^{24}x^{30}$

Wait! There’s a contradiction — I made a mistake in the first method.

Wait: $(8z^4x^5)^3 = 512z^{12}x^{15}$, and $(z^4x^5)^3 = z^{12}x^{15}$, so multiplying gives:

$512z^{12}x^{15} \cdot z^{12}x^{15} = 512z^{24}x^{30}$

But earlier I said $8 \cdot (z^4x^5)^6$, which would be $8 \cdot z^{24}x^{30}$ → wrong!

Mistake: $(8z^4x^5)^3 = 8^3 (z^4x^5)^3 = 512(z^4x^5)^3$, not $8(z^4x^5)^3$

So correct way:

$(8z^4x^5)^3 = 512z^{12}x^{15}$

$(z^4x^5)^3 = z^{12}x^{15}$

So total: $512z^{12}x^{15} \cdot z^{12}x^{15} = 512z^{24}x^{30}$

✔ Answer: $512z^{24}x^{30}$

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11) $(z^25x^3)^2$

This is $(5x^3z^2)^2$

- $5^2 = 25$
- $(x^3)^2 = x^6$
- $(z^2)^2 = z^4$

✔ Answer: $25x^6z^4$

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12) $6(9yx^4)^3$

First: $(9yx^4)^3 = 9^3 \cdot y^3 \cdot (x^4)^3 = 729y^3x^{12}$

Now multiply by 6: $6 \cdot 729y^3x^{12} = 4374y^3x^{12}$

✔ Answer: $4374y^3x^{12}$

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13) $(2zy^5)^2 \cdot (z^5y^5)^3$

First: $(2zy^5)^2 = 2^2 \cdot z^2 \cdot (y^5)^2 = 4z^2y^{10}$

Second: $(z^5y^5)^3 = z^{15}y^{15}$

Now multiply: $4z^2y^{10} \cdot z^{15}y^{15} = 4z^{2+15}y^{10+15} = 4z^{17}y^{25}$

✔ Answer: $4z^{17}y^{25}$

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14) $(z5x^4)^2$

This is $(5x^4z)^2$

- $5^2 = 25$
- $(x^4)^2 = x^8$
- $z^2$

✔ Answer: $25x^8z^2$

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15) $(5zy^2)^2 \cdot (zy^4)^3$

First: $(5zy^2)^2 = 25z^2y^4$

Second: $(zy^4)^3 = z^3y^{12}$

Multiply: $25z^2y^4 \cdot z^3y^{12} = 25z^{5}y^{16}$

✔ Answer: $25z^5y^{16}$

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16) $(6y^2z^2)^2 \cdot (y^3)^2$

First: $(6y^2z^2)^2 = 36y^4z^4$

Second: $(y^3)^2 = y^6$

Multiply: $36y^4z^4 \cdot y^6 = 36y^{10}z^4$

✔ Answer: $36y^{10}z^4$

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17) $7(8y^2z^3)^2$

First: $(8y^2z^3)^2 = 64y^4z^6$

Now multiply by 7: $7 \cdot 64y^4z^6 = 448y^4z^6$

✔ Answer: $448y^4z^6$

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18) $(3zy^3)^3 \cdot (z^3)^3$

First: $(3zy^3)^3 = 27z^3y^9$

Second: $(z^3)^3 = z^9$

Multiply: $27z^3y^9 \cdot z^9 = 27z^{12}y^9$

✔ Answer: $27z^{12}y^9$

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✔ Final Answers:

| Problem | Answer |
|--------|--------|
| 1 | $2y^{10}z^4$ |
| 2 | $8y^3z^{18}$ |
| 3 | $8y^{18}z^9$ |
| 4 | $49x^8y^4$ |
| 5 | $9x^4z^{10}$ |
| 6 | $9y^8z^6$ |
| 7 | $256x^3y^{15}$ |
| 8 | $324x^6y^{10}$ |
| 9 | $2048x^3z^6$ |
| 10 | $512z^{24}x^{30}$ |
| 11 | $25x^6z^4$ |
| 12 | $4374y^3x^{12}$ |
| 13 | $4z^{17}y^{25}$ |
| 14 | $25x^8z^2$ |
| 15 | $25z^5y^{16}$ |
| 16 | $36y^{10}z^4$ |
| 17 | $448y^4z^6$ |
| 18 | $27z^{12}y^9$ |

Let me know if you'd like this formatted as a printable answer key or need explanations for any specific problem!