Solving Exponential Problems

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Now, we can "cancel" any instance of a factor that appears in both the numerator and denominator. Recall that we were able to find equivalent fractions by multiplying (or dividing) both the numerator and denominator by a particular value-this is equivalent to multiplying or dividing by one.Later in the course, we will consider fractional exponents.(As it turns out, fractional exponents obey the same rules as integer exponents, but the precise meaning of a fraction will be made clear later on.) Although exponents may at times seem like an obscure or less than practical mathematical tool, they have numerous important and practical applications.Note also the usefulness of these rules of exponents in part d: multiplying 150 twenty (or twenty-six) times is a tough proposition, and even most calculators cannot provide an exact product.The use of exponents, however, allows us to have an Notice that the expanded form has four factors of 2 and four factors of 3.Exponential functions often involve the rate of increase or decrease of something.When it's a rate of increase, you have an exponential growth function!We can see that the exponent of the answer is the difference between that of the numerator and that of the denominator (again, all have the same base).Let's generalize the rule: Notice that the expression in parentheses has three factors, and we must multiply this expression four times.This expression is the same as the following (multiplication is commutative, so we can rearrange the factors in the product).If you consider division of exponential expressions, you may notice that the rule seems to indicate that we can end up with negative exponents.

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