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