EDGE Math 8

Laws of Exponents

Consider the following examples that use the definition of a positive integer exponent:

b^{2} \cdot b^{3} = (b \cdot b) (b \cdot b \cdot b)

= b \cdot b \cdot b \cdot b \cdot b = b^{5}

x^{3} \cdot x^{4} = (x \cdot x \cdot x) (x \cdot x \cdot x \cdot x)

= x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x = x^{7}

a^{2} \cdot a^{2} = (a \cdot a) (a \cdot a)

= a \cdot a \cdot a \cdot a = a^{4}

These examples illustrate the first law of positive integer exponents.

Product Rule
If $a$ is a real number and $m$ and $n$ are positive integers, then

a^{m} \cdot a^{n} = a^{m + n} .

This rule means that when you are multiplying two exponential expressions with the same base, you just need to add the exponents. For example, to multiply $r^{9} \cdot r^{5},$ add their exponents; that is,

r^{9} \cdot r^{5} = r^{9 + 5} = r^{14} .