On occasion, it is not possible to obtain the result of a power directly, as is the case of a fractional exponent power. For this reason, I am going to explain you how to solve this type of potencies. I’m going to tell you about the exponential form of the roots, that is, how to pass from root to power and vice versa.
On the other hand, passing a root to its power form can be very useful to you at a given time if you are working with powers.
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Powers with fractional exponent. How to solve a number elevated to a fraction
To solve a fractional exponent power, you must pass from power to root form according to this formula:
When you have a power with fractional exponent, it is the same as if you had a root, where the denominator of the exponent is the index of the root and the numerator of the exponent is the exponent of the radicand (content of the root).
Let’s see a few examples of how to solve powers with fractional exponent to make it clearer:
We apply the previous formula: 2 becomes the radicando (root content), 4 becomes the root index and 7 remains as exponent of 2:
In this case, the root doesn’t have a whole solution so it stays that way. We have simply passed it from power to root to express the result in another way, but it has not served to obtain the result of the root.
The same thing will not happen with this other power:
Directly you can’t calculate the result, but by transforming it to root if:
The 2 becomes the index of the root and the 1 to elevate to the 4. As you know the index of the square roots is not written even when the exponents are 1 either, so keep it in mind. I just put them so you would know.
When the fractional exponent has a 1 as numerator, no exponent will appear in the radicand:
Negative Fractional Exponents
Of course, you must be very careful with the minus signs. You can have them in the exponent or in the base and when they are in the base, they can be affected by the exponent or not.
Let’s take a slower look at each of these cases.
We begin with the case of when we have a negative sign in the exponent, as for example:
The first thing you have to do is pass the exponent from negative to positive and this is done by passing the power to the denominator.
Once you have the power in the denominator, you have a positive exponent in the form of a fraction, which you can transform into a root, but keep in mind that it remains in the denominator:
Other case is that you have the negative sign in the base of the power:
In this case, the minus sign is outside the power, that is, it is equivalent to the power being multiplied by -1.
Therefore, the minus sign is maintained and the power is passed to root form, to obtain its result:
If the minus sign is enclosed in parentheses, then it is affected by the exponent:
In this case, the minus sign becomes part of the radicand as it was at the base of the power.
As a root with even index there is no real solution.
Remember with only the roots with odd index results when the radicand is negative.
Potencies with fractional exponent with bases formed by several factors or terms
The same happens when the minus sign precedes the power, that is, it is not affected by the exponent and it is as if multiplied by -1, the power with fractional exponent can be multiplied by a number or a variable, as for example:
This case is very easily solved, since we only have to maintain the 3 and pass to root form the power to which it multiplies, leaving:
It may also be the case that in the power base you have several numbers and variables. In order for all of them to be part of the base, they must be enclosed in parentheses.
Let’s see an example:
As you can see, all factors become part of the radicand, just as they were at the base of the power.
Then the 7 goes on to elevate all factors, so they must again be enclosed in parentheses
The base can also be formed by several terms such as this power:
As in the previous case, the base, which is the one enclosed in parentheses, becomes the radicand, which in turn is elevated to 3, which was the numerator of the fractional exponent. The 5 becomes the index of the root.
Everything you’ve just seen can be mixed with each other, to the point where you can find powers with bases formed by several terms, which in turn have several factors and are elevated to a negative fraction.
You should only go step by step, solving only one problem at each step.