# How do you simplify 3^5*3^4?

Feb 13, 2016

${3}^{5} \cdot {3}^{4} = {3}^{9}$

#### Explanation:

Using the property ${a}^{n} \cdot {a}^{m} = {a}^{n + m}$:

${3}^{5} \cdot {3}^{4} = {3}^{5 + 4} = {3}^{9}$

To gain some intuition as to why this property works, let's try expanding the exponent into multiplication:

${a}^{n} \cdot {a}^{m} = \left(a \cdot a \cdot a \cdot \ldots \cdot a\right) \left(a \cdot a \cdot a \cdot \ldots \cdot a\right)$ where the first parentheses contain $n$ $a$'s and the second parentheses contain
$m$ $a$'s. Thus, there are $n + m$ total $a$'s being multiplied, or ${a}^{n + m}$

The above is a simple way of seeing why this property works for positive integer exponents, however the property does hold for all exponents (including negative numbers, fractions, irrational numbers, and so on).