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Performing operations on exponents

When two terms have variables that are raised by exponents, and we multiply these terms together, we do not multiply the exponents. Instead, we add them together. Thus, if we have x^2 multiplied by x^3, we come out with x^5. When we see two terms whose variables have exponents divided by one another, we do not divide the exponents themselves. Instead, we subtract the one from the other. Thus, if we have x^6/x^4, we end up with x^2.

We have seen before that when we add two terms whose variables have exponents, we do not multiply the exponents themselves, but instead add them. But there are cases in which we do multiply exponents by one another. (x^2)^6 would be such an example. Our result would be x^12. A similar principle is operative in cases like (xy)^9. The distributive property is used in this case, and we get the result x^9y^9. Likewise, if this exponent is applied to a fraction, such as (x/y)^9, we get x^9/y^9. The process of multiplying rational exponents is the same as that for multiplying rational numbers or fractions. The numerators of both are multiplied by one another and the denominators are multiplied by one another. So 2x^3/5 times 3x^1/2 = 6x^3/10.