Multiplication in Algebraic Expressions worksheet - Free Printable

Let's solve each problem step by step, following the rules of multiplying algebraic expressions:

Rules to Remember:

- Multiply coefficients (numbers).
- Add exponents when multiplying like bases.
- Handle signs carefully (negative × negative = positive, etc.).
- Write answers in the format: `coefficient*variable^exponent`, e.g., `-1/2*x^2*y^3`.

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1.) $ a^2 \times 3a^4b $

- Coefficients: $ 1 \times 3 = 3 $
- Variables: $ a^2 \times a^4 = a^{2+4} = a^6 $
- Include $ b $: $ b^1 $
- Result: $ 3a^6b $

✔ Answer: `3*a^6*b`

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2.) $ \frac{2}{3}p \times 15p^3 $

- Coefficients: $ \frac{2}{3} \times 15 = \frac{30}{3} = 10 $
- Variables: $ p \times p^3 = p^{1+3} = p^4 $
- Result: $ 10p^4 $

✔ Answer: `10*p^4`

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3.) $ (-5mn^2) \times 14m^2n^7p $

- Coefficients: $ -5 \times 14 = -70 $
- $ m \times m^2 = m^{1+2} = m^3 $
- $ n^2 \times n^7 = n^{9} $
- $ p $ remains $ p $
- Result: $ -70m^3n^9p $

✔ Answer: `-70*m^3*n^9*p`

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4.) $ (-8a^3bc^4) \times \left(-\frac{3}{4}a^2b^3c\right) $

- Coefficients: $ -8 \times -\frac{3}{4} = +6 $
- $ a^3 \times a^2 = a^{5} $
- $ b \times b^3 = b^4 $
- $ c^4 \times c = c^5 $
- Result: $ 6a^5b^4c^5 $

✔ Answer: `6*a^5*b^4*c^5`

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5.) $ (-2x^2y^2z) \times (-9x^4y^2z^5) \times (-3x^3yz^4) $

- Coefficients: $ -2 \times -9 = 18 $, then $ 18 \times -3 = -54 $
- $ x^2 \times x^4 \times x^3 = x^{2+4+3} = x^9 $
- $ y^2 \times y^2 \times y = y^{2+2+1} = y^5 $
- $ z \times z^5 \times z^4 = z^{1+5+4} = z^{10} $
- Result: $ -54x^9y^5z^{10} $

✔ Answer: `-54*x^9*y^5*z^10`

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6.) $ 2\frac{1}{2}j^3k^2 \times (-1\frac{1}{2}j^4k^3) $

Convert mixed numbers:
- $ 2\frac{1}{2} = \frac{5}{2} $
- $ -1\frac{1}{2} = -\frac{3}{2} $

Now multiply:
- Coefficients: $ \frac{5}{2} \times -\frac{3}{2} = -\frac{15}{4} $
- $ j^3 \times j^4 = j^7 $
- $ k^2 \times k^3 = k^5 $
- Result: $ -\frac{15}{4}j^7k^5 $

✔ Answer: `-15/4*j^7*k^5`

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7.) $ 4\frac{1}{7}p^5q^2r^3 \times (-1\frac{5}{58}p^3qr^2) \times (-4pq^4r^5s) $

Convert mixed numbers:
- $ 4\frac{1}{7} = \frac{29}{7} $
- $ -1\frac{5}{58} = -\frac{63}{58} $

Multiply coefficients:
- $ \frac{29}{7} \times -\frac{63}{58} = -\frac{29 \times 63}{7 \times 58} $
- Simplify: $ \frac{29}{58} = \frac{1}{2} $, $ \frac{63}{7} = 9 $
- So: $ -\frac{1}{2} \times 9 = -\frac{9}{2} $
- Now multiply by $ -4 $: $ -\frac{9}{2} \times -4 = +18 $

Now variables:
- $ p^5 \times p^3 \times p = p^{5+3+1} = p^9 $
- $ q^2 \times q \times q^4 = q^{2+1+4} = q^7 $
- $ r^3 \times r^2 \times r^5 = r^{10} $
- $ s $ appears once

Result: $ 18p^9q^7r^{10}s $

✔ Answer: `18*p^9*q^7*r^10*s`

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8.) $ (-6\frac{2}{3}l^3m^4n^{-5}) \times (-2\frac{1}{2}l^2m^2n^{12}) \times (-1\frac{1}{2}m^3n^{-4}p^2) \times (-3\frac{1}{11}) $

First, convert all mixed numbers:

- $ -6\frac{2}{3} = -\frac{20}{3} $
- $ -2\frac{1}{2} = -\frac{5}{2} $
- $ -1\frac{1}{2} = -\frac{3}{2} $
- $ -3\frac{1}{11} = -\frac{34}{11} $

Now multiply coefficients:
- $ -\frac{20}{3} \times -\frac{5}{2} = +\frac{100}{6} = \frac{50}{3} $
- $ \frac{50}{3} \times -\frac{3}{2} = -\frac{150}{6} = -25 $
- $ -25 \times -\frac{34}{11} = +\frac{850}{11} $

Now variables:
- $ l^3 \times l^2 = l^5 $
- $ m^4 \times m^2 \times m^3 = m^{9} $
- $ n^{-5} \times n^{12} \times n^{-4} = n^{-5+12-4} = n^3 $
- $ p^2 $ is only in last term
- No other variables

So result: $ \frac{850}{11} l^5 m^9 n^3 p^2 $

✔ Answer: `850/11*l^5*m^9*n^3*p^2`

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9.) $ 4\frac{1}{2}w^5x^4yz^2 \times \left(-\frac{5}{13}w^2xyz^3\right) \times 3\frac{1}{2}w^{-3}x^{-5}y^4z $

Convert mixed numbers:
- $ 4\frac{1}{2} = \frac{9}{2} $
- $ 3\frac{1}{2} = \frac{7}{2} $

Coefficients:
- $ \frac{9}{2} \times -\frac{5}{13} = -\frac{45}{26} $
- $ -\frac{45}{26} \times \frac{7}{2} = -\frac{315}{52} $

Variables:
- $ w^5 \times w^2 \times w^{-3} = w^{5+2-3} = w^4 $
- $ x^4 \times x \times x^{-5} = x^{4+1-5} = x^0 = 1 $
- $ y \times y \times y^4 = y^{1+1+4} = y^6 $
- $ z^2 \times z^3 \times z = z^{6} $

So result: $ -\frac{315}{52} w^4 y^6 z^6 $

✔ Answer: `-315/52*w^4*y^6*z^6`

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10.) $ (-2\frac{2}{3}a^5b^9cd^{18}) \times (-1\frac{1}{17}a^{-2}b^{-4}c^{-3}d^{-13}) \times 3a^3bc^6d^2 $

Convert mixed numbers:
- $ -2\frac{2}{3} = -\frac{8}{3} $
- $ -1\frac{1}{17} = -\frac{18}{17} $

Coefficients:
- $ -\frac{8}{3} \times -\frac{18}{17} = +\frac{144}{51} = \frac{48}{17} $
- $ \frac{48}{17} \times 3 = \frac{144}{17} $

Variables:
- $ a^5 \times a^{-2} \times a^3 = a^{5-2+3} = a^6 $
- $ b^9 \times b^{-4} \times b = b^{9-4+1} = b^6 $
- $ c \times c^{-3} \times c^6 = c^{1-3+6} = c^4 $
- $ d^{18} \times d^{-13} \times d^2 = d^{18-13+2} = d^7 $

Result: $ \frac{144}{17} a^6 b^6 c^4 d^7 $

✔ Answer: `144/17*a^6*b^6*c^4*d^7`

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✔ Final Answers:

1. `3*a^6*b`
2. `10*p^4`
3. `-70*m^3*n^9*p`
4. `6*a^5*b^4*c^5`
5. `-54*x^9*y^5*z^10`
6. `-15/4*j^7*k^5`
7. `18*p^9*q^7*r^10*s`
8. `850/11*l^5*m^9*n^3*p^2`
9. `-315/52*w^4*y^6*z^6`
10. `144/17*a^6*b^6*c^4*d^7`

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