Introduction to exponents

​ ​You already know that multiplication is repeated addition, right? In the same way we can use exponential expressions to write repeated multiplications. 3x3x3x3x3 is three multiplied by itself five times. We can write it this way: three raised to the power of five to save some space. The larger number to the left is the base.

Here, the base is 3. This is the number that is going to be multiplied by itself. The smaller number, above the base is the power. it tells you how many times you should multiply the base by itself. Together the base and the power make an exponential expression.

You read it as three to the power of five. ​ ​Exponential expressions with the powers of two and three have their own special names. We call them squared or cubed, four square or five cubed for example. This comes from geometry. The area of a square is after all, one side multiplied by itself two times. And the volume of a cube is one side times itself three times.

Squares and cubes are simply different ways of expressing this. ​ ​When you see an exponential expression, it always means the same thing. Y raised to the power of X means Y multiplied by itself X times. So Y raised to the power of X is completely different from Y times X. ​ ​Leon forgot the code to his lock. Each number is between zero and nine, or ten different alternatives. How many different codes does Leon have to try before he exhausted all combinations?

You can write this problem as an exponential expression. For each of the three digits, you have ten alternatives. That is, 10x10x10, or ten raised to the power of three, which is 1000. There are 1000 different possible combinations for Leon's lock. ​ ​Maria borrowed 5000 kronor to buy a bicycle. She pays 5% interest per year.

If she doesn't pay back anything for five years, how much will she owe? After one year, she owes the original amount, times 1.05. And for every extra year, the amount owed including the interest is multiplied by 1.05. 1.05 multiplied by itself five times is the same as 1.05 to the power of five. So Maria's debt will be 5000 times 1.05, raised to the power of five.

To pay back the loan, Maria now has to pay 6381 kronor. ​ ​Exponential expressions consist of a base and a power. The base is multiplied by itself as many times as the power says. Exponential expressions with the power of two can be thought of as squares, and they're often called base squared.