Exponent rules

Prerequisites
All four operations for the rational numbers (+,-,*,/)

What is an exponent?
There are rules of exponents. But first, what are exponents? Exponents are repeated multiplication. For example: 5^3=5*5*5.

Product of Powers
The product of powers rule states that given any three numbers, say 3,4 and 5, (3^4)*(3^5)=3^(4+5). This is simple to show. In this example, 3^4 means 3*3*3*3 and 3^5 means 3*3*3*3*3. Multiplying both together, we get (3*3*3*3)*(3*3*3*3*3). But wait! That's just 4+5 or 9 threes multiplied together, or 3^(4+5).

Power to a power
This rule says that given any three numbers, say 2,3 and 4, (2^3)^4=2^(3*4). In this example, 2^3=2*2*2 and (2^3)^4=(2^3)*(2^3)*(2*3)*(2^3). Putting it all together, we get: (2*2*2)*(2*2*2)*(2*2*2)*(2*2*2). That's just 3*4 or 12 twos multiplied together, or 2^(3*4).

Quotient of Powers
This rule states that given any three numbers, say 2,3 and 5, (5^3)/(5^2)=5^(3-2). In this example, 5^3=5*5*5 and 5^2=5*5. Using this knowledge, we can show that (5^3)/(5^2)=5*(5*5)/(5*5), or 5, which is just (3-2) or one five multiplied with itself, or 5^(3-2)

What does this mean?
First off, say we wanted to know 5^(1/3). We know that, based on the rules, that (5^(1/3))^3=5^(1/3*3) or 5. Whatever 5^(1/3) is, when raised to the power three, it's five. We can thus also write it as $$\sqrt[3]{5}$$.

Also, what does 3^-1 mean? Well, it's 3^(1-2) or 3^1/3^2, 3/(3*3). Divide both sides to get 1/3.

The next level up is defined in a certain way. Take 3^^4. It means 3^3^3^3. The powers go from right to left, rather than left to right because doing it left to right is redundant ((((3^3)^3)^3)=3^(3*3*3), but 3^(3^(3^3)) actually brings something new.)