Product of powers property can be used to find the product of monomials.
Product of Powers Property :
The product of two powers with the same base equals that base raised to the sum of the exponents.
If x is any nonzero real number and m and n are integers, then
x^{m} ⋅ x^{n} = x^{m+n}
Example 1 :
Multiply.
(4y^{2})(5y^{3})
Solution :
= (4y^{2})(5y^{3})
Group factors with like bases together.
= (4 ⋅ 5)(y^{2} ⋅ y^{3})
Use the Product of Powers Property.
= 20y^{2 + }^{3}
= 20y^{5}
Example 2 :
Find the product of the following monomials.
-5x and 3y
Solution :
= -5x ⋅ 3y
= -(5 ⋅ 3)xy
= -15xy
Example 3 :
Find the product of the following monomials.
(³⁄₇)x^{5} and (¹⁴⁄₉)x^{2}
Solution :
= (³⁄₇)x^{5} ⋅ (¹⁴⁄₉)x^{2}
Group factors with like bases together.
= (³⁄₇ ⋅ ¹⁴⁄₉)(x^{5 }⋅ x^{2})
Use the Product of Powers Property.
= (¹⁄₁ ⋅ ⅔)(x^{5 + }^{2})
= (⅔)x^{7}
Example 4 :
Find the product of the following monomials.
xyz and x^{2}yz
Solution :
= xyz ⋅ x^{2}yz
Group factors with like bases together.
= (x ⋅ x^{2})(y ⋅ y)(z ⋅ z)
Use the Product of Powers Property.
= (x^{1 + }^{2})(y^{1 + 1})(z^{1 + 1})
= (x^{3})(y^{2})(z^{2})
= x^{3}y^{2}z^{2}
Example 5 :
Find the product of the following monomials
x^{3}y^{5} and xy^{2}
Solution :
= x^{3}y^{5} ⋅ xy^{2}
Group factors with like bases together.
= (x^{3} ⋅ x)(y^{5} ⋅ y^{2})
Use the Product of Powers Property.
= (x^{3 + 1})(y^{5 + 2})
= (x^{4})(y^{7})
= x^{4}y^{7}
Example 6 :
Find the product of the following monomials.
a^{2}b^{2}c^{3 }and abc^{2}
Solution :
= a^{2}b^{2}c^{3} ⋅ abc^{2}
Group factors with like bases together.
= (a^{2} ⋅ a)(b^{2} ⋅ b)(c^{3} ⋅ c^{2})
Use the Product of Powers Property.
= (a^{2 + 1})(b^{2 + 1})(c^{3 + }^{2})
= (a^{3})(b^{3})(c^{5})
= a^{3}b^{3}c^{5}
Example 7 :
Find the product of the following monomials.
4ab, 3a^{2}b^{2 }and 2a^{3}b^{3}
Solution :
= 4ab ⋅ 3a^{2}b^{2}⋅ 2a^{3} b^{3}
Group factors with like bases together.
= (4 ⋅ 3 ⋅ 2)(a ⋅ a^{2 }⋅ a^{3})(b ⋅ b^{2} ⋅ b^{3})
Use the Product of Powers Property.
= (24)(a^{1 + 2 + 3})(b^{1 + 2 + 3})
= 24a^{6}b^{6}
Example 8 :
Find the product of the following monomials.
-5pq, 2p^{2}q and -4q^{5}
Solution :
= -5pq ⋅ 2p^{2}q ⋅ -4q^{5}
Group factors with like bases together.
= -(5 ⋅ 2 ⋅ -4)(p ⋅ p^{2})(q ⋅ q^{ }⋅ q^{5})
Use the Product of Powers Property.
= -(-40)p^{1 + 2}q^{1 + 1 + 5}
= 40p^{3}q^{7}
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