Mathematical Problem Solving/Exponents and Logarithms

Integer Exponents
It is often useful to repeatedly add a number to itself. For example, one might want to run around a track of length 400 five times. We write $$5 \cdot 400$$ instead of $$400 + 400 + 400 + 400 + 400$$. Multiplication can be said to be repeated addition: i.e. $$x \cdot y = x + x + \cdots + x$$ where $$x$$ is added five times. Just as we have shorthand for repeated addition, we can describe repeated multiplication. We write $$2^5$$ instead of $$2 \cdot 2 \cdot 2 \cdot 2 \cdot 2$$. In fact we will now define exponentiation for integer exponents. If $$n$$ is an integer, then
 * $$x^n = x \cdot x \cdots x$$

When referring to the previous expression we would say $$x$$ is the base and $$n$$ is the exponent or power.

Examples
Evaluate $$2^5 \cdot 2^6$$.

Solution: The first term is the product of five 2's and six 2's, giving us 11 2's.

Evaluate $$\frac{3^{15}}{3^{12}}$$.