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

Problem: Factor the given monomials into their prime factors. Do not use exponents.

We will solve each problem step by step, breaking down the coefficients into their prime factors and keeping the variables as they are.

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#### 1) \( 77x^2 \)

- Coefficient: \( 77 \)
- Prime factorization of \( 77 \):
- \( 77 = 7 \times 11 \)
- Variables: \( x^2 \) (no further factoring needed)

- Final Answer: \( 7 \cdot 11 \cdot x \cdot x \)

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#### 2) \( 60xy^2 \)

- Coefficient: \( 60 \)
- Prime factorization of \( 60 \):
- \( 60 = 2 \times 30 \)
- \( 30 = 2 \times 15 \)
- \( 15 = 3 \times 5 \)
- So, \( 60 = 2 \times 2 \times 3 \times 5 \)
- Variables: \( xy^2 \) (no further factoring needed)

- Final Answer: \( 2 \cdot 2 \cdot 3 \cdot 5 \cdot x \cdot y \cdot y \)

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#### 3) \( 62qz \)

- Coefficient: \( 62 \)
- Prime factorization of \( 62 \):
- \( 62 = 2 \times 31 \)
- Variables: \( qz \) (no further factoring needed)

- Final Answer: \( 2 \cdot 31 \cdot q \cdot z \)

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#### 4) \( 64c^2w^3 \)

- Coefficient: \( 64 \)
- Prime factorization of \( 64 \):
- \( 64 = 2 \times 32 \)
- \( 32 = 2 \times 16 \)
- \( 16 = 2 \times 8 \)
- \( 8 = 2 \times 4 \)
- \( 4 = 2 \times 2 \)
- So, \( 64 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \)
- Variables: \( c^2w^3 \) (no further factoring needed)

- Final Answer: \( 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot c \cdot c \cdot w \cdot w \cdot w \)

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#### 5) \( 78rg^3 \)

- Coefficient: \( 78 \)
- Prime factorization of \( 78 \):
- \( 78 = 2 \times 39 \)
- \( 39 = 3 \times 13 \)
- So, \( 78 = 2 \times 3 \times 13 \)
- Variables: \( rg^3 \) (no further factoring needed)

- Final Answer: \( 2 \cdot 3 \cdot 13 \cdot r \cdot g \cdot g \cdot g \)

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#### 6) \( 78y^3h \)

- Coefficient: \( 78 \)
- Prime factorization of \( 78 \):
- \( 78 = 2 \times 39 \)
- \( 39 = 3 \times 13 \)
- So, \( 78 = 2 \times 3 \times 13 \)
- Variables: \( y^3h \) (no further factoring needed)

- Final Answer: \( 2 \cdot 3 \cdot 13 \cdot y \cdot y \cdot y \cdot h \)

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#### 7) \( 78s^2n \)

- Coefficient: \( 78 \)
- Prime factorization of \( 78 \):
- \( 78 = 2 \times 39 \)
- \( 39 = 3 \times 13 \)
- So, \( 78 = 2 \times 3 \times 13 \)
- Variables: \( s^2n \) (no further factoring needed)

- Final Answer: \( 2 \cdot 3 \cdot 13 \cdot s \cdot s \cdot n \)

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#### 8) \( 74m^2h^2 \)

- Coefficient: \( 74 \)
- Prime factorization of \( 74 \):
- \( 74 = 2 \times 37 \)
- Variables: \( m^2h^2 \) (no further factoring needed)

- Final Answer: \( 2 \cdot 37 \cdot m \cdot m \cdot h \cdot h \)

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#### 9) \( 48p^3y^2 \)

- Coefficient: \( 48 \)
- Prime factorization of \( 48 \):
- \( 48 = 2 \times 24 \)
- \( 24 = 2 \times 12 \)
- \( 12 = 2 \times 6 \)
- \( 6 = 2 \times 3 \)
- So, \( 48 = 2 \times 2 \times 2 \times 2 \times 3 \)
- Variables: \( p^3y^2 \) (no further factoring needed)

- Final Answer: \( 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot p \cdot p \cdot p \cdot y \cdot y \)

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#### 10) \( 54g \)

- Coefficient: \( 54 \)
- Prime factorization of \( 54 \):
- \( 54 = 2 \times 27 \)
- \( 27 = 3 \times 9 \)
- \( 9 = 3 \times 3 \)
- So, \( 54 = 2 \times 3 \times 3 \times 3 \)
- Variables: \( g \) (no further factoring needed)

- Final Answer: \( 2 \cdot 3 \cdot 3 \cdot 3 \cdot g \)

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Final Answers:

1. \( 7 \cdot 11 \cdot x \cdot x \)
2. \( 2 \cdot 2 \cdot 3 \cdot 5 \cdot x \cdot y \cdot y \)
3. \( 2 \cdot 31 \cdot q \cdot z \)
4. \( 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot c \cdot c \cdot w \cdot w \cdot w \)
5. \( 2 \cdot 3 \cdot 13 \cdot r \cdot g \cdot g \cdot g \)
6. \( 2 \cdot 3 \cdot 13 \cdot y \cdot y \cdot y \cdot h \)
7. \( 2 \cdot 3 \cdot 13 \cdot s \cdot s \cdot n \)
8. \( 2 \cdot 37 \cdot m \cdot m \cdot h \cdot h \)
9. \( 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot p \cdot p \cdot p \cdot y \cdot y \)
10. \( 2 \cdot 3 \cdot 3 \cdot 3 \cdot g \)

\boxed{
\begin{aligned}
1. & \ 7 \cdot 11 \cdot x \cdot x \\
2. & \ 2 \cdot 2 \cdot 3 \cdot 5 \cdot x \cdot y \cdot y \\
3. & \ 2 \cdot 31 \cdot q \cdot z \\
4. & \ 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot c \cdot c \cdot w \cdot w \cdot w \\
5. & \ 2 \cdot 3 \cdot 13 \cdot r \cdot g \cdot g \cdot g \\
6. & \ 2 \cdot 3 \cdot 13 \cdot y \cdot y \cdot y \cdot h \\
7. & \ 2 \cdot 3 \cdot 13 \cdot s \cdot s \cdot n \\
8. & \ 2 \cdot 37 \cdot m \cdot m \cdot h \cdot h \\
9. & \ 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot p \cdot p \cdot p \cdot y \cdot y \\
10. & \ 2 \cdot 3 \cdot 3 \cdot 3 \cdot g
\end{aligned}
}