In the product [[$ 4^2 \cdot 4^3 $]], the base number of both powers is the same. The expression is called the product of powers with the same base.

[[$ 4^2 \cdot 4^3 = \underbrace{4 \cdot 4}_{2 \text{ kpl}} \cdot \underbrace{4 \cdot 4 \cdot 4}_{+3 \text{ kpl}} = 4^5 = 1024 $]]

Example 1

Simplify the powers.

a)	[[$ a^2 \cdot a^4 = a^{2+4} = a^6 $]]
b)	[[$ x \cdot x^2 \cdot x^3 = x^{1+2+3} = x^6 $]]
c)	[[$ a^3 \cdot a^2 \cdot b \cdot b^6= a^{3+2} \cdot b^{1+6} = a^5b^7 $]]	Only powers powers with the same base can be combined.

Multiplications of powers with the same base often involve other factors that can be combined separately. If there are variables or letters in the product, the multiplication sign is omitted between the numeric value and the variable.

Example 2

Simplify the powers.

a)	[[$ (-2) \cdot (-2)^2 = (-2)^{1+2} = (-2)^3 = -8 $]]
b)	[[$ -3 \cdot 3^3 = -3^{1+3} = -3^4 = -81 $]]
c)	[[$ 2x^2 \cdot x^6 = 2x^{2+6} = 2x^8 $]]
d)	[[$ 3a^4 \cdot (-2a^3) = 3 \cdot (-2) \cdot a^4 \cdot a^3 = -6a^{4+3} = -6a^7 $]]	The numbers are multiplied and the exponents are added together.