In the case of positive exponents, we easily multiply the number base by itself, but what happens when we have negative numbers as exponents? A negative exponent is defined as the multiplicative inverse of the base, raised to the power which is opposite to the given power.
In simple words, we write the reciprocal of the number and then solve it like positive exponents. We know that the exponent of a number tells us how many times we should multiply the base. For example, consider 8 2 , 8 is the base, and 2 is the exponent. A negative exponent tells us, how many times we have to multiply the reciprocal of the base.
Consider the 8 -2 , here, the base is 8 and we have a negative exponent Here are a few examples which express negative exponents with variables and numbers. Observe the table to see how the number is written in its reciprocal form and how the sign of the powers changes. We have a set of rules or laws for negative exponents which make the process of simplification easy. Given below are the basic rules for solving negative exponents. A negative exponent takes us to the inverse of the number.
This is how negative exponents change the numbers to fractions. Instead of flipping twice, I noted that all the powers were negative, and moved the outer power onto the inner ones; since "minus times minus is plus", I ended up with all positive powers. Note: While this second solution would be a faster way of getting the exercise done, "faster" doesn't mean "more right". Either way is fine. Since exponents indicate multiplication, and since order doesn't matter in multiplication, there will often be more than one sequence of steps that will lead to a valid simplification of a given exercise of this type.
Don't worry if the steps in your homework look quite different from the steps in a classmate's homework. As long as your steps were correct, you should both end up with the same answer in the end. You can use the Mathway widget below to practice simplifying expressions with negative exponents. Try the entered exercise, or type in your own exercise.
Then click the button to compare your answer to Mathway's. Or skip the widget and continue with the lesson. Please accept "preferences" cookies in order to enable this widget. Click here to be taken directly to the Mathway site, if you'd like to check out their software or get further info. By the way, now that you know about negative exponents, you can understand the logic behind the "anything to the power zero" rule:.
Why is this so? There are various explanations. One might be stated as "because that's how the rules work out. At each stage, with each stage having a power than was one less than what came before, the simplified value was equal to the previous value, divided by 3.
Again, just move the number to the denominator of a fraction to make the exponent positive. One way you can rewrite the question we're given is the following:. Multiplying in that -1 will turn the equation back into what it was originally. However, keeping the -1 outside helps us work with the negative exponent a little easier and allows us to illustrate what's happening.
So moving on from the above, we can continue solving with the negative exponent as we did before. If you ever see a negative exponent on the top of a fraction, you know that if you flip it to the bottom, it'll become positive. The same actually works for negative exponents on the bottom. If you move it to the numerator, its exponent also becomes positive.
With that in mind, let's work through the question. Our first step is just to flip the numerator and denominator to get rid of all the negatives in the exponents. Then solve as usual with the power rule. Here's a good place to take a look at comparing negative and positive exponents and seeing how they behave on a graph. Back to Course Index.
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Why is a value with a negative exponent equal to the multiplicative inverse but with a positive exponent? But there is more to it than that. Further mathematical developments, which you have not seen yet, confirm these choices.
If it didn't work out this way, we would suspect that something was wrong somewhere. And in fact it has often happened that mathematicians have tried defining something one way, and then later developments revealed that the definition was not the right one, and it had to be revised.
Here, though, that did not happen. Much of this is copied from my answer to a similar question earlier. The arithmetic function which is the inverse of multiplication is division. Try it - take a number and multiply it by 2, then divide it by 2. You'll come up with the original answer.
The OP doesn't seem to understand what a negative exponent means. Assuming this was not actually the question asked for homework.
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