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Hacking exponents

The value of an exponential function is the number produced when a number is multiplied by itself n times.

For example, 12^2 = 144 or 11^2 = 121.

Fair enough, but how do I multiply, divide, add, and subtract exponents in their unresolved form?

Suppose we were asked to find the result of 12^2 times 12^6. Well we could solve each exponent individually or we could apply a rule to make our lives simpler. This brings us to our first rule.

When multiplying exponents with a common base, we add the exponents together and raise the base to the the new sum. So in our example 12^2 times 12^6 becomes 12^8 because 6+2 = 8. So 12^8 = 12*12*12*12*12*12*12*12 = 429,981,696 which coincidentally is the number of times I have to tell my nephew to pick up his socks each day :)

Suppose we were asked to find the result of 12^3 / 12^2. Our second rule is the opposite of the first. We subtract the exponents, and raise the common base to the value of the difference.

Now 12^3 / 12^2 becomes 12^1 because 3-2 = 1.

The answer is 12, coincidentally the number of days of Christmas, the number of months in a year, and the number of eggs in a dozen.

If, for instance, an exponent is itself raised to another power, the rules change slightly. Typically, an exponent is written within parentheses. Our next rule requires us to multiply the inner exponent by the outer exponent. So (3^2)^3 = 3^6 , because 3 times 2 = 6 and the answer is 729. Coincidentally, this is the number of days between two birthdays during a non-leap and leap year.



Thomas D.

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