# MULTIPLYING  MONOMIALS

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)(x⋅ x2)

Use the Product of Powers Property.

=  (¹⁄₁ ⋅ )(x5 + 2)

=  ()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, 3a2band 2a3b3

Solution :

=  4ab ⋅ 3a2b2⋅ 2a3 b3

Group factors with like bases together.

=  (4 ⋅ 3 ⋅ 2)(a ⋅ a⋅ 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  2p2⋅ -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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