# Negative and fractional exponents

Categories: exponentials

We have seen how to calculate the exponential function *a ^{x}* for any integer value of x of 2 or greater.

Now we will extend this to cover:

- Rules for one and zero powers.
- Negative exponent rule.
- Fractional exponent rule.

## Assumptions

We will make some further assumptions:

- All values are real. It is possible to calculate exponentials involving complex numbers, we won't cover it here.
- The base is not negative. This is because exponential equations for negative bases can yield complex number results for certain values of
*x*. - The base is not zero. This is because exponential equations for base zero are undefined results for certain values of
*x*.

This means that we will assume the base is positive and the exponent is any real number.

## Rules for one and zero powers.

The exponentials *a ^{2}*,

*a*.

^{3}*a*form a series where each item is equal to the previous term multiplied by

^{4}*a*. Every time we multiply by

*a*, the power increases by 1.

We can also generate the series in reverse order like this:

Starting from *a ^{3}*, each time we divide the item by

*a*, the exponent decreases by 1.

This allows us to find that the value of:

- $a^1$. This is
*a*. This makes sense because one multiplied*a*once is just*a*. - $a^0$. Again, by the original definition, this is one multiplied by
*a*zero times, which is one.

## Special bases 1 and 0

Powers of one, such as 1, 1 × 1, and 1 × 1 × 1, are all equal to 1. We can say that 1 raised to any power is 1.

Similarly, 0, 0 × 0, and 0 × 0 × 0 are all equal to 0. So 0 raised to any power is 0.

But there is a special case. Previously we saw that any number raised to the power 0 is equal to 1. But we also know that 0 raised to any (non-negative) power is 0. So what is *0 ^{0}*?

We say that this value is undefined. This is similar to the case of 0 divided by 0, which is also undefined.

As we said earlier, we are only considering positive bases in this article, and since 0 is not positive we do not need to worry about the special case.

## Negative exponent rule.

Suppose we carry on with the previous technique and keep dividing by *a* and reducing the exponent by 1:

This tells us that *a ^{-1}* is equal to the reciprocal of

*a*.

If we use the definition we expect that *a ^{-1}* is one multiplied by

*a*minus one times. What does that mean? Well if we interpret that as meaning we do the

*opposite*of multiplying, one time, it means that the result should be one divided by

*a*. Which is exactly what it is.

In general *a ^{-p}* is the reciprocal of

*a*.

^{p}## Fractional exponent rule.

Can we raise *a* to the power one-half? And what would that mean? Here is an illustration:

We set *y* equal to *a* to the power one half (whatever that might be).

Using the power to a power rule, *a* raised to the power 1/2, then raised the power 2, is just equal to *a*.

This means that *y ^{2}* to equal

*a*, making

*y*equal to the square root of

*a*.

Incidentally, this is an example of why we are only considering positive bases. If *a* was -1, we would need to calculate the square root of -1, which has no real number value.

But the same logic, *a* to the power one-third is the cube root of *a*:

This can be extended to include any rational number as a power:

Here we use the fact that raising *a* to the power of 3/2 is equivalent to raising *a* to the power of 3 followed by raising it to the power of 1/2. In that way, any rational power can be calculated by raising *a* to an integer power and then taking an integer root.

The article Exponential function for irrational values of x shows this can be extended to cover irrational valued powers.

## See also

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