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## Exponents Worksheets

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What are exponents?

Sometimes, mathematics advances when an old idea is communicated in a new way. A new notation that makes concepts clearer can actually lead to fresh insights. This is what exponents did.

When a number is multiplied by itself, the expression can be written (for example) like this:

2 x 2 x 2 x 2; or, it can be abbreviated like this: 24.

The little 4 is the exponent, and it means multiply the base (in this case 2) by itself 4 times.

You should know that exponents are also called powers, so that in the above example, you could also say, "two to the fourth power".

 How are exponents used in the real world? Exponents do not have to be whole numbers. Although the meaning is harder to understand, an expression like 21.5 is perfectly valid. (The exponent here is 1.5) This is important because many events in the real world follow a pattern of some base number raised to a fractional exponent. Whenever this pattern appears, it is called a 'power law'. An important example is earthquakes. Although any particular earthquake cannot be predicted, the number and size of earthquakes worldwide in a year follows a pattern. The pattern is given by a power law: 10(8 - M) where the exponent is (8 - M). M stands for magnitude, which is a measure of an earthquake's power. Using this formula and putting a 6 in for magnitude gives 10(8 - 6) and since 8 - 6 = 2, the formula tells us that there will be 102 earthquakes of magnitude 6 expected in a year. 102 is the same as 10 x 10, or 100. The important thing is that as the strength (magnitude) of earthquakes go down, the number of them goes up. In fact, there will be (on average) 1000 more of magnitude 5 than magnitude 6, and 10,000 more of magnitude 4. A basic problem using exponents. Exponents can be added together when numbers are multiplied together. If you have 53 times 52 this is the same as saying (5 x 5 x 5) times (5 x 5), which is the same as 5 multiplied by itself five times total, or 55. The exponents end up adding together: 53 x 52 equals 53+2. This works any time the base (in this example, the base was 5) is the same and the numbers are multiplied together. Who invented exponents? The use of exponents to mean "multiply by itself this many times" can be found in a book from the 1400's by Nicolas Chuquet, a French mathematician. The actual word was coined in the 1500's by Michael Stifel. He wrote in Latin, and exo meaning 'out of', along with ponenere which means 'place', gave us the word 'exponent' - to move out of place. This just refers to the way the little number is lifted above the line of text - it is moved out of place, above the line. An interesting fact about exponents. Any number or expression with an exponent of zero is equal to one. 60 = 9990 = X0 = 1 Knowing this, see if you get the following math joke, which usually appears on t-shirts: [on the front of the t-shirt]