# Powers and exponents

Calculate:${12}^{6}*{12}^{3}$

## Powers and exponents

​ ​You know that we use exponential expressions to express repeated multiplications and that you multiply the base by itself as many times as the power shows, right? Now we are going to do calculations with exponential expressions. Let's begin with multiplication. Multiplying two exponential expressions with the same base is easy. Look here.

Three squared times three cubed is three times three times three times three times three. This is three multiplied by itself five times. Look at the powers. You get to the answer by adding the powers. Two plus three is five.

Three squared times three cubed equals three to the power of five. ​ ​Division of exponential expressions works the same way. A repeated multiplication divided by a repeated multiplication can be simplified. Remove the factors in the numerator and denominator as many above the line as below it. Since we have two series of repeating factors, we can remove the shorter string entirely. Which makes it four times four, divided by one.

Or simply, four times four. Or, four to the power of two. The power in the denominator shows how many factors you can remove from the number in the numerator. So, you just subtract the power in the denominator from the power in the numerator. Multiplication and division of exponential expressions with the same base is the same is adding or subtracting the powers.

With multiplication, X to the power of A times X to the power of B, is equal to X to the power of A plus B. And with division, X to the power of A divided by X to the power of B, is equal to X to the power of A minus B. ​ ​These are the first and second rules of exponents. Now it gets a little bit complicated. What happens when you take an exponential expression and raise it to a power again? Take two for example.

Raise it to the power of three and then square the results. Two to the power of three to the power of two. Write it out as a repeated multiplication and you might be able to see what happens. Raising an exponential expression to a power is the same as multiplying the powers. This is the third rule of exponents.

X to the power of A to the power of B is equal to X to the power of A times B. ​ ​Here is an interesting example. You know that a number divided by itself equals one, right? Now you also know that the second rule of exponents says that division of exponential expressions with the same base is the same as subtracting the denominators' power from the numerators. X to the power of four divided by X to the power of four must be one since we have the same number in the numerator and the denominator. But it must also be equal to X to the power of zero since it is X to the power of four minus four, which is X to the power of zero.

This means that X to the power of zero equals one. In fact any number raised to the power of zero is equal to one. This is the fourth rule of exponents. Exponential expressions contain a base and a power. The base is multiplied by itself as many times as the power says.

When you multiply exponent expressions with the same base you add the powers, when you divide exponential expressions with the same base you simply subtract the powers. A power of a power, an exponential expression raised to a power is calculated by multiplying the powers. Any positive number raised to the power of zero equals.