How many exponent laws are there

If you memorize these three definitions, you can work everything else out by combining them and by counting:. Let me make good on that promise, by showing you how all the other laws of exponents come from just the three definitions above. Suppose you have x 5 x 6 ; how do you simplify that?

Why x 11? Five x factors from x 5 , and six x factors from x 6 , makes eleven x factors total. Can you see that whenever you multiply any two powers of the same base, you end up with a number of factors equal to the total of the two powers? In other words, when the bases are the same , you find the new power by just adding the exponents :. The rule above works only when multiplying powers of the same base.

For instance,. Except in one case: If the bases are different but the exponents are the same , then you can combine them. And it works for any common power of two different bases:. What about dividing? Remember that dividing is just multiplying by 1-over-something. So all the laws of division are really just laws of multiplication.

Well, there are several ways to work it out. However you slice it, you come to the same answer: for division with like bases you subtract exponents , just as for multiplication of like bases you add exponents:. As that example illustrates, you can combine like exponents even when the bases are different:. Use the rule you already know for dividing:. Solved Examples on Exponents. Practice Test on Exponents. Worksheet on Exponents. Didn't find what you were looking for? Or want to know more information about Math Only Math.

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When multiplying two terms it does not matter if they are like terms , multiply the coefficients together to get the new coefficient. Then, one at a time, add the powers of each variable to make the new powers.

If you multiplied. When a term that includes variables with exponents is raised to another power, raise the coefficient to that power and multiply each existing power by the second power to find the new exponent. Anything raised to the first power stays the same. Anything raised to the power of 0 becomes the number 1. It doesn't matter how complicated or big the term is. To divide when you have the same variable in the numerator and denominator, and the larger exponent is on top, subtract the bottom exponent from the top exponent to calculate the value of the exponent of the variable on top.

Then, eliminate the bottom variable. Reduce any coefficients like a fraction. To divide when you have the same variable in the numerator and denominator, and the larger exponent is on the bottom, subtract the top exponent from the bottom exponent to calculate the new exponential value on the bottom. Then, erase the variable from the numerator and reduce any coefficients like a fraction. If there are no variables left on top, leave a 1.

To eliminate negative exponents, put the term under 1 and change the exponent so that the exponent is positive. For example,.