Before you understand this concept, you are advised to read:
What are Monomials ?
How to Multiply Integers ?
What are like terms ?
What are Unlike terms ?
While multiplying two monomials you will find the following situations:
Multiply two monomials having like terms
Example 1: Multiply (5a) and (10a)
Multiply two monomials having unlike terms
Example 2: Multiply (3p) and (7q)
Multiply two negative monomials and both have like terms
Example 3: Multiply (-4d) and (-2d)
Multiply two negative monomials having unlike terms
Example 4: Multiply (-8y) and (-7z)
Multiply two monomials where one monomial is positive & other is negative and both have like terms
Example 5: Multiply (-6s) and (2s)
Multiply two monomials where one monomial is positive & other is negative and have unlike terms
Example 6: Multiply (-3p) and (7q)
Multiply two monomials having like terms
Example 1: Multiply 5a and 10a
Solution: This proceeds as:
Write terms in the multiplication expression and we get:
5a X 10a
Multiply constants and variable separately, as shown below:
(5 X 10 ) X (a X a)
Multiply constants as we multiply numbers & we get:
(50) X (a X a)
Multiply variables as per the Law of exponent, am X an = am+n, we get:
50 X a 2
Multiply constant and variable and we get:
50a2
Hence, (5a X 10a) = (50a2)
Multiply two monomials having unlike terms
Example 2: Multiply (3q) and (7p)
Solution: This proceeds as:
Write terms in the multiplication expression and we get:
3q X 7p
Multiply constants and variable separately, as shown below:
(3 X 7 ) X (q X p)
Multiply constants as we multiply numbers & we get:
(21) X (q X p)
Multiply variables & we get:
21 X qp
Multiply constant and variables and we get:
21qp
Or we write it as: 21pq (because variables are arranged in alphabetical order)
Hence, (3p X 7q) = (21pq)
Multiply two negative monomials and both have like terms
Example 3: Multiply (-4d) and (-2d)
Solution: This proceeds as:
Write terms in the multiplication expression and we get:
-4d X -2d
Multiply constants and variable separately, as shown below:
(-4 X -2 ) X (d X d)
Multiply constants as we multiply negative integers & we get:
(8) X (d X d)
Multiply variables as per the Law of exponent, am X an = am+n, we get:
8 X d2
Multiply constant and variables and we get:
8d2
Hence, (-4d X -2d) = (8d2)
Multiply two negative monomials having unlike terms
Example 4: Multiply (-8y) and (-7z)
Solution: This proceeds as:
Write terms in the multiplication expression and we get:
-8y X -7z
Multiply constants and variable separately, as shown below:
(-8 X -7) X (y X z)
Multiply constants as we multiply negative integers & we get:
56 X (y X z)
Multiply variables & we get:
56 X yz
Multiply constant and variables and we get:
56yz
Hence, (-8y X -7z) = (56yz)
Multiply two monomials where one monomial is positive & other is negative and both have like terms
Example 5: Multiply (-6s) and (2s)
Solution: This proceeds as:
Write terms in the multiplication expression and we get:
-6s X 2s
Multiply constants and variable separately, as shown below:
(-6 X 2 ) X (s X s)
Multiply constants as we multiply positive and negative integers & we get:
(-12) X (s X s)
Multiply variables as per the Law of exponent, am X an = am+n, we get:
-12 X s2
Multiply constant and variables and we get:
(-12s2)
Hence, (-6s X 2s) = (-12s2)
Multiply two monomials where one monomial is positive & other is negative and have unlike terms
Example 6: Multiply (-3p) and 7q
Solution: This proceeds as:
Write terms in the multiplication expression and we get:
-3p X 7q
Multiply constants and variable separately, as shown below:
(-3 X 7 ) X (p X q)
Multiply constants as we multiply positive and negative integers & we get:
(-21) X (p X q)
Multiply variables & we get:
-21 X pq
Multiply constant and variables and we get:
-21pq
Hence, (-3p X 7q) = (-21pq)
By now you must have understood how to multiply two monomials under different situations, below you can find some more examples where more than two monomials are multiplied:
Example 7: Multiply (2p)(-3q)(2p)
Solution: This proceeds as:
Write terms in the multiplication expression and we get:
2p X -3q X 2p
Multiply constants and variable separately, as shown below:
(2 X -3 X 2) X (p X q X p)
Multiply constants as we multiply integers & we get:
(-12) X (p X q X p)
Rearrange variable & we get:
(-12) X (p X p X q)
Multiply Variables and we get:
-12 X p2q
Multiply constant and variables and we get:
-12p2q
Hence, (2p X -3q X 2p) = (-12p2q)
Example 8: Multiply (-x)(-3xy)(6y)
Solution: This proceeds as:
Write terms in the multiplication expression and we get:
-x X-3xy X 6y
Multiply constants and variable separately, as shown below:
(-1 X -3 X 6) X (x X xy X y)
Multiply constants as we multiply integers & we get:
(18) X (x X xy X y)
Rearrange variable & we get:
(18) X (x X x X y X y)
Multiply Variables and we get:
18 X x2y2
Multiply constant and variables and we get:
18 X x2y2
Hence, (-x X-3xy X 6y) = (18 X x2y2)
Example 9: Multiply (-2a)(-3c)(-3d)(4b)
Solution: This proceeds as:
Write terms in the multiplication expression and we get:
-2a X -3c X -3d X 4b
Multiply constants and variable separately, as shown below:
(-2 X -3 X -3 X 4) X (a X c X d X b)
Multiply constants as we multiply integers & we get:
(-72) X (a X c X d X b)
Multiply Variables and we get:
(-72) X acdb
Multiply constant and variables and we get:
-72acdb
Or we write it as: -72abcd (because variables are arranged in alphabetical order)
Hence, (-2a X -3c X -3d X 4b) = (-72abcd)
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