Operation on Algebraic Expression worksheet - Free Printable

Problem Explanation and Solution

The worksheet is divided into two parts:
1. Part A: Adding and Subtracting — Simplify algebraic expressions by combining like terms.
2. Part B: Multiplying and Dividing — Perform multiplication and division of algebraic expressions.

Let's solve each part step by step.

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Part A: Adding and Subtracting

#### Instructions: Combine like terms to simplify each expression.

1. 3x + 8x
- Combine the coefficients of \( x \):
\[
3x + 8x = (3 + 8)x = 11x
\]
- Answer: \( 11x \)

2. 2y + 3y − y
- Combine the coefficients of \( y \):
\[
2y + 3y - y = (2 + 3 - 1)y = 4y
\]
- Answer: \( 4y \)

3. 13m + 2m + 6m
- Combine the coefficients of \( m \):
\[
13m + 2m + 6m = (13 + 2 + 6)m = 21m
\]
- Answer: \( 21m \)

4. 6p + 4p − 3p
- Combine the coefficients of \( p \):
\[
6p + 4p - 3p = (6 + 4 - 3)p = 7p
\]
- Answer: \( 7p \)

5. 10y − 11y + y
- Combine the coefficients of \( y \):
\[
10y - 11y + y = (10 - 11 + 1)y = 0y = 0
\]
- Answer: \( 0 \)

6. −2k + 5k + 11k − 8k
- Combine the coefficients of \( k \):
\[
-2k + 5k + 11k - 8k = (-2 + 5 + 11 - 8)k = 6k
\]
- Answer: \( 6k \)

7. 3 + 6p + 8 − 2p
- Combine the constant terms and the coefficients of \( p \):
\[
3 + 6p + 8 - 2p = (3 + 8) + (6p - 2p) = 11 + 4p
\]
- Answer: \( 11 + 4p \)

8. 6y³ + 6y + 3y³ − 8y
- Combine the coefficients of \( y^3 \) and \( y \):
\[
6y^3 + 6y + 3y^3 - 8y = (6 + 3)y^3 + (6 - 8)y = 9y^3 - 2y
\]
- Answer: \( 9y^3 - 2y \)

9. −2y − 3y − 4y + y
- Combine the coefficients of \( y \):
\[
-2y - 3y - 4y + y = (-2 - 3 - 4 + 1)y = -8y
\]
- Answer: \( -8y \)

10. 4k² + 3 − 2k² − 11
- Combine the coefficients of \( k^2 \) and the constant terms:
\[
4k^2 + 3 - 2k^2 - 11 = (4 - 2)k^2 + (3 - 11) = 2k^2 - 8
\]
- Answer: \( 2k^2 - 8 \)

11. x² − 3x − x + 8
- Combine the coefficients of \( x \) and keep \( x^2 \) as it is:
\[
x^2 - 3x - x + 8 = x^2 + (-3 - 1)x + 8 = x^2 - 4x + 8
\]
- Answer: \( x^2 - 4x + 8 \)

12. p² − 3m + p² + 2m
- Combine the coefficients of \( p^2 \) and \( m \):
\[
p^2 - 3m + p^2 + 2m = (1 + 1)p^2 + (-3 + 2)m = 2p^2 - m
\]
- Answer: \( 2p^2 - m \)

13. 4d + 17 + 5d − 22
- Combine the coefficients of \( d \) and the constant terms:
\[
4d + 17 + 5d - 22 = (4 + 5)d + (17 - 22) = 9d - 5
\]
- Answer: \( 9d - 5 \)

14. x² + 3x − 3x² − 6x
- Combine the coefficients of \( x^2 \) and \( x \):
\[
x^2 + 3x - 3x^2 - 6x = (1 - 3)x^2 + (3 - 6)x = -2x^2 - 3x
\]
- Answer: \( -2x^2 - 3x \)

15. 6x² + 8x − 3x² + 12x
- Combine the coefficients of \( x^2 \) and \( x \):
\[
6x^2 + 8x - 3x^2 + 12x = (6 - 3)x^2 + (8 + 12)x = 3x^2 + 20x
\]
- Answer: \( 3x^2 + 20x \)

16. 2m − 6 + 4m + 11 − m
- Combine the coefficients of \( m \) and the constant terms:
\[
2m - 6 + 4m + 11 - m = (2 + 4 - 1)m + (-6 + 11) = 5m + 5
\]
- Answer: \( 5m + 5 \)

17. 3ab + 7ba + 2b
- Note that \( ab = ba \), so combine the coefficients of \( ab \) and keep \( 2b \) as it is:
\[
3ab + 7ba + 2b = (3 + 7)ab + 2b = 10ab + 2b
\]
- Answer: \( 10ab + 2b \)

18. 4mn + 6n − 3nm − 2n
- Note that \( mn = nm \), so combine the coefficients of \( mn \) and \( n \):
\[
4mn + 6n - 3nm - 2n = (4 - 3)mn + (6 - 2)n = mn + 4n
\]
- Answer: \( mn + 4n \)

19. −7pq + 9ef − 3pq − 2ef
- Combine the coefficients of \( pq \) and \( ef \):
\[
-7pq + 9ef - 3pq - 2ef = (-7 - 3)pq + (9 - 2)ef = -10pq + 7ef
\]
- Answer: \( -10pq + 7ef \)

20. 2xy³ + 3x³y − 6xy³ − 3x³y
- Combine the coefficients of \( xy^3 \) and \( x^3y \):
\[
2xy^3 + 3x^3y - 6xy^3 - 3x^3y = (2 - 6)xy^3 + (3 - 3)x^3y = -4xy^3 + 0x^3y = -4xy^3
\]
- Answer: \( -4xy^3 \)

21. 16m²y² − 3m²y² − 12y²m²
- Combine the coefficients of \( m^2y^2 \):
\[
16m^2y^2 - 3m^2y^2 - 12m^2y^2 = (16 - 3 - 12)m^2y^2 = 1m^2y^2 = m^2y^2
\]
- Answer: \( m^2y^2 \)

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Part B: Multiplying and Dividing

#### Instructions: Perform multiplication and division of algebraic expressions.

1. 2 × 3a
- Multiply the constants and keep the variable:
\[
2 \times 3a = 6a
\]
- Answer: \( 6a \)

2. 4b × 2b
- Multiply the constants and add the exponents of \( b \):
\[
4b \times 2b = (4 \times 2)b^{1+1} = 8b^2
\]
- Answer: \( 8b^2 \)

3. 6x × (−4y)
- Multiply the constants and keep the variables:
\[
6x \times (-4y) = (6 \times -4)xy = -24xy
\]
- Answer: \( -24xy \)

4. 3p × (−2p) × 5
- Multiply the constants and add the exponents of \( p \):
\[
3p \times (-2p) \times 5 = (3 \times -2 \times 5)p^{1+1} = -30p^2
\]
- Answer: \( -30p^2 \)

5. 24x ÷ x
- Divide the constants and subtract the exponents of \( x \):
\[
24x \div x = 24x^{1-1} = 24
\]
- Answer: \( 24 \)

6. 16x² ÷ 8x
- Divide the constants and subtract the exponents of \( x \):
\[
16x^2 \div 8x = \frac{16}{8}x^{2-1} = 2x
\]
- Answer: \( 2x \)

7. 15a²b² ÷ 3ab
- Divide the constants and subtract the exponents of \( a \) and \( b \):
\[
15a^2b^2 \div 3ab = \frac{15}{3}a^{2-1}b^{2-1} = 5ab
\]
- Answer: \( 5ab \)

8. 24ab² × 3b
- Multiply the constants and add the exponents of \( b \):
\[
24ab^2 \times 3b = (24 \times 3)a b^{2+1} = 72ab^3
\]
- Answer: \( 72ab^3 \)

9. −4 × 3d × 2d
- Multiply the constants and add the exponents of \( d \):
\[
-4 \times 3d \times 2d = (-4 \times 3 \times 2)d^{1+1} = -24d^2
\]
- Answer: \( -24d^2 \)

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Final Answers

#### Part A: Adding and Subtracting
1. \( 11x \)
2. \( 4y \)
3. \( 21m \)
4. \( 7p \)
5. \( 0 \)
6. \( 6k \)
7. \( 11 + 4p \)
8. \( 9y^3 - 2y \)
9. \( -8y \)
10. \( 2k^2 - 8 \)
11. \( x^2 - 4x + 8 \)
12. \( 2p^2 - m \)
13. \( 9d - 5 \)
14. \( -2x^2 - 3x \)
15. \( 3x^2 + 20x \)
16. \( 5m + 5 \)
17. \( 10ab + 2b \)
18. \( mn + 4n \)
19. \( -10pq + 7ef \)
20. \( -4xy^3 \)
21. \( m^2y^2 \)

#### Part B: Multiplying and Dividing
1. \( 6a \)
2. \( 8b^2 \)
3. \( -24xy \)
4. \( -30p^2 \)
5. \( 24 \)
6. \( 2x \)
7. \( 5ab \)
8. \( 72ab^3 \)
9. \( -24d^2 \)

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Boxed Final Answer

\[
\boxed{
\text{Part A: }
1. 11x,
2. 4y,
3. 21m,
4. 7p,
5. 0,
6. 6k,
7. 11 + 4p,
8. 9y^3 - 2y,
9. -8y,
10. 2k^2 - 8,
11. x^2 - 4x + 8,
12. 2p^2 - m,
13. 9d - 5,
14. -2x^2 - 3x,
15. 3x^2 + 20x,
16. 5m + 5,
17. 10ab + 2b,
18. mn + 4n,
19. -10pq + 7ef,
20. -4xy^3,
21. m^2y^2
}
\]
\[
\boxed{
\text{Part B: }
1. 6a,
2. 8b^2,
3. -24xy,
4. -30p^2,
5. 24,
6. 2x,
7. 5ab,
8. 72ab^3,
9. -24d^2
}
\]