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
xm ⋅ xn = xm+n
Example 1 :
Multiply.
(4y2)(5y3)
Solution :
= (4y2)(5y3)
Group factors with like bases together.
= (4 ⋅ 5)(y2 ⋅ y3)
Use the Product of Powers Property.
= 20y2 + 3
= 20y5
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)x5 and (¹⁴⁄₉)x2
Solution :
= (³⁄₇)x5 ⋅ (¹⁴⁄₉)x2
Group factors with like bases together.
= (³⁄₇ ⋅ ¹⁴⁄₉)(x5 ⋅ x2)
Use the Product of Powers Property.
= (¹⁄₁ ⋅ ⅔)(x5 + 2)
= (⅔)x7
Example 4 :
Find the product of the following monomials.
xyz and x2yz
Solution :
= xyz ⋅ x2yz
Group factors with like bases together.
= (x ⋅ x2)(y ⋅ y)(z ⋅ z)
Use the Product of Powers Property.
= (x1 + 2)(y1 + 1)(z1 + 1)
= (x3)(y2)(z2)
= x3y2z2
Example 5 :
Find the product of the following monomials
x3y5 and xy2
Solution :
= x3y5 ⋅ xy2
Group factors with like bases together.
= (x3 ⋅ x)(y5 ⋅ y2)
Use the Product of Powers Property.
= (x3 + 1)(y5 + 2)
= (x4)(y7)
= x4y7
Example 6 :
Find the product of the following monomials.
a2b2c3 and abc2
Solution :
= a2b2c3 ⋅ abc2
Group factors with like bases together.
= (a2 ⋅ a)(b2 ⋅ b)(c3 ⋅ c2)
Use the Product of Powers Property.
= (a2 + 1)(b2 + 1)(c3 + 2)
= (a3)(b3)(c5)
= a3b3c5
Example 7 :
Find the product of the following monomials.
4ab, 3a2b2 and 2a3b3
Solution :
= 4ab ⋅ 3a2b2⋅ 2a3 b3
Group factors with like bases together.
= (4 ⋅ 3 ⋅ 2)(a ⋅ a2 ⋅ a3)(b ⋅ b2 ⋅ b3)
Use the Product of Powers Property.
= (24)(a1 + 2 + 3)(b1 + 2 + 3)
= 24a6b6
Example 8 :
Find the product of the following monomials.
-5pq, 2p2q and -4q5
Solution :
= -5pq ⋅ 2p2q ⋅ -4q5
Group factors with like bases together.
= -(5 ⋅ 2 ⋅ -4)(p ⋅ p2)(q ⋅ q ⋅ q5)
Use the Product of Powers Property.
= -(-40)p1 + 2q1 + 1 + 5
= 40p3q7
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