Formula and Execution
Form of Fraction Exponent: a = xn/d
Notice the form written in the row above. The number which is in place of x is the base that is required to be substituted in this section.
The exponent here is in the form of a fraction, i.e., n/d, where n is the numerator, and d is the denominator. The numerator of the fraction is supposed to be plugged in here.
The denominator of the fraction, i.e., the number in place of d will be plugged in here.
Fractional Exponent (a)
This section is the final result section. When you put all your values in the right sections, look for your answers here!
Fraction Exponent Calculator
Welcome to our online fraction exponent calculator! I know it’s a bad day. Fractions and that too, in the exponents, are gimmicking at your face. But don’t worry, we are here at your service!
All our online tools will always be there to make sure that your bad day turns to a good one. Just like every other calculator on our list, you simply have to substitute the value in the right section and look up for the results. Our algorithms are designed in a way that keeps you away from the strain of calculating things manually and give instant results.
How to Use Fraction Exponent Calculator
Look up to our formula and execution table above to know which value is supposed to be substituted in which column. Once all your values are in the right sections, you get quick results.
The good news is, our calculator works in any way you want it to. Need numerator? No need to carry out complicated algebra to derive the formula for finding numerator. Just plug in the other three values, the numerator will automatically show up.
The same goes for all other values. Whether you have to calculate denominator, base, or simply the fractional exponent, knock us up!
A screenshot is attached here. I worked out all straight. Substituted base, numerator, denominator, and got required fractional exponent result. But the calculator is too flexible to accept any three values and yield the fourth one. Smart enough, right? Play with any values for fun!
Concept of Fractional Exponent
The word fraction has the ability to annoy anyone. It sounds so intimidating that you keep on procrastinating your homework. But one should have the discipline of not being afraid of things in mathematics when they look difficult. Especially in mathematics, when things look difficult, they are very, very powerful. And this is the case with the fractional exponents, that's exponents that are fractions.
Once you understand them, they are as easy as they could possibly be! Besides, once you master them, you will be able to master a monster called logarithm, which will bring all kinds of nice tools to us.
Before we start with explaining you fractional exponent, we consider it to be better if you read the content on our exponent calculator. When foundations are strong, the building is definitely stronger.
For now, in this article, we only recall the laws of exponent, and a few generalized forms, whose proofs can be seen in the exponent calculator.
Laws of Exponent
- xm xn = xm+n
- xm/xn = xm-n
- (xm)n = xmn
- (xn)(yn) = (xy)n
- xn/yn = (x/y)n
- x-n = 1/xn or xn = 1/x-n
- 0(any negative number) = undefined
- 0(any positive number) = 0
- 1(any number) = 1
Now, to begin with, fractional exponents, aka rational exponents, are nothing but another way to express radicals like square roots, cube roots, fourth roots, nth roots, or radicals with powers on them like the square of the cube root of something.
Let us look into a few examples
x1/2 * x1/2 = x1/2+1/2 = x
x1/3 * x1/3 * x1/3 = x1/3+1/3+1/3 = x
x1/4 * x1/4 * x1/4 * x1/4 = x1/4+1/4+1/4+1/4 = x
Do you notice any pattern in the above examples? We have simply applied the law of exponent: xm xn = xm+n, on the above examples. We already know how to add up fractions. Working with fractions in exponents is no different than working with fractions normally. Recall the concept of square roots, cube roots, and nth roots of numbers. Roots are the operations which, when multiplied to itself, the number of roots time, gives the original number.
In the cases we considered above, we consider the original number to be x. In every case, when we multiply the fractional exponents by itself, we get x. It means that these operations are nothing but square root, cube root, and fourth root of x.
In radical notation, we can write the above forms as,
√x * √x = x
∛x * ∛x * ∛x = x
And similarly, we can write for fourth root, fifth root, and so on as well. So we say, exponent of ½ is a square root, exponent of 1/3 is cube root, and in general, exponent of 1/n is nth root. Mathematically, we express this as,
x1/n = n√x
Now let us take this a step further. We take an nth root and raise it to some power. We consider the example ∛x and raise it to the power of 2, i.e., (∛x)2. We can certainly write this in the form of exponential notation, and it would be, (x1/3)2. Now by the law of exponent: (xm)n = xmn, this expression can be represented as,
- (x1/3)2 = x2/3
We consider another example
- (4√x)5 = (x1/4)5 = (x5/4)
So, in general,
xm/n = (n√x)m
The left-hand side of the above expression is known as the rational/fractional exponent notation, and the right-hand side is the radical notation of the same thing.
We can use either form to simplify our problems. We show this with the help of a few examples.
By using the law of exponent to write the power, we get
(251/2)3 or (253)1/2
Both of the above forms are correct and equal to one another. But we choose to go with the form that is easier to get to the result. Since we know that the exponent ½ is equal to the square root, so in the first case, we take the square root of 25 and then cube it. While in the second case, we first take the cube of 25, then take the square root of the result. Either way, we will get the same answer, but of course, taking the square root first and then cubing it is a lot simpler.
So we chose to work with the form (251/2)3.
- (251/2)3 = (√25)3 = (5)3 = 125
Using the law of exponent, we can write this as,
Either (81/3)2 or (82)1/3
In this case, working either way, is pretty simple, because the cube root of 8 is 2, and its square would be 4. While with the second case, the square of 8 is 64, whose cube root is 4. We get the same results, no matter which way we walk. Just remember to choose the simpler path!
The exponents can be natural numbers, fractions, zero, or even negative numbers. I assume that you already know how to deal with negative exponents. What if we have negative with a fraction? Well, don’t panic. It’s no big deal.
Consider the example (1000)-2/3.
We have already solved the examples above but without the negative sign. We follow the same pattern and write the above number as (10001/3)-2. 1000 to the power 1/3 is the cube root of 1000, which is equal to 10. Then we simply raise 10 to the power of negative 2 and get our required results.
- (1000)-2/3 = (10001/3)-2 = (∛1000)-2 = (10)-2 = 1/102 = 1/100
It wasn’t difficult, right? We can also write the above expression as 1/(1000)2/3 or 1/(∛1000)2. This will yield the exact results. Either reciprocate it before taking root or after taking root, it puts no impact on the result. In general, we write as,
x-m/n = 1/(n√x)m
To summarize what we studied above, we divide the fractional exponents into two categories and state the general forms of each category.
- Fractional/rational exponents with 1 in the numerator with positive and negative exponents.
x1/n = n√x
x-1/n = 1/n√x
- Fractional/rational exponents with a numerator other than 1 with positive and negative exponents.
xm/n = (n√x)m
x-m/n = 1/(n√x)m
The Graph of Fractional Exponents
Below is the image that illustrates the graph of fractional exponents with numerators 1. The graphs plotted shows not only the square root, fourth root, and eighth root, but also shows the graphs of the square, fourth power, and eighth power. Notice how they are the inverses of one another; therefore, they are a reflection of each other.
The Derivative of Fractional Exponent
The derivative of a fractional exponent is calculated exactly as the derivative of any function with integers in their powers is calculated.
We know that the power rule for the derivative is,
d/dx (x)n = n(x)n-1
What if we have a fraction in the exponent like x2/5? The same power rule would still be applicable. If we generalize it, we write it as,
d/dx (x)m/n = (m/n) (x)[(m/n) – 1]
Example: Find derivative of x2/5
d/dx (x)2/5 = (2/5) (x)2/5 – 1 = (2/5) (x)-3/5
If we make the power positive, we write the above expression as,
So, the same power rule can be applied to the fractional exponents as well. Now try out calculating the derivative of the function 2/√x.
How can you calculate fractional exponents in a calculator?
Every calculator has different operations located at different places. We describe here two methods for two different views of calculator.
- Look up in your calculator for the sign “^”.
- Once you have found this sign, simply type the number, then press the “^” button.
- Write your exponents in parenthesis and press the “=” sign to get the results.
Example: Let us consider the number 72/3. The way the number would look like when you type in the calculator is 7^(2/3).
- Type the number first, which is the base.
- Now, lookup for the button in your calculator that looks like “x2”
- Press this button, and a square box will appear on the exponent of your number.
- Now, lookup for the fraction button that looks like “/”.
- Press the fraction button. Type in your numerator in the top box. To move to the bottom box, press the down arrow key. Now type in your denominator.
- Your fractional exponent would look like 72/3.
- Press “=” sign, and you get the instant results!
Note: While working with our calculator, simply substitute the right number in the right section, and get precise results without the need of pressing the “=” sign. Besides, our calculator is flexible enough to not just calculate fractional exponents, but also the base, numerator, or denominator of the fraction without going through the headache of carrying out difficult algebra! Substitute any three values and get the fourth one instantly. Awesome, right?
When do you flip or reciprocate the numbers in fractional exponents?
When the exponents are negative, we reciprocate or flip them to make the exponent positive. Like x-2/3 will be written as 1/x2/3 to make the exponents positive.
What if we have a negative fraction exponent in the denominator?
Just like we flip x-1/n to 1/x1/n, similarly, if the negative fraction exponent is in the denominator, we will flip it and bring it to the numerator. 1/x-1/n would be written as x1/n.
What happens if the base of the fractional exponent is negative?
There are two cases in this regard:
In the first one, we first solve the fractional exponent, then multiply it by the negative sign. So, our result, in this case, would be negative.
But in the second case, things get complicated. If we are working on the real values only, then the answer to this number would be invalid. But if we extend our scope to complex numbers, then our answer would be imaginary based on the fact that roots of negative numbers are imaginary since √-1 = i, which is the imaginary number known as an iota.
Can we have irrational numbers as exponents?
Yes, just like rational exponents, we can have irrational numbers like e, π, √2, etc. Since this calculator is limited to rational, aka fractional exponents only, so we would consider the discussion of irrational exponents beyond the scope of this article. But you can easily solve these in your regular scientific calculator with the power option.
How do we simplify problems with fractional exponents?
Solving fractional exponents is no rocket science. The first and foremost step is to get away with the fear of fractions. Once your fear has vanished into thin air, the next step is to solve the fraction exponents just like we solve integer exponents. The laws of exponents will be applicable depending upon whether the bases are the same or the powers are the same. Once the laws are applied, then the next step is to carry out algebra between the exponents. That would exactly be how we normally solve the fractions when they are not being used in the exponents.
Example: Solve x-1/4 * x2/3
Since the bases are the same, we apply the same base law and add up the powers.
- x-1/4 + 2/3 = x5/12 (simply solving the fractions by taking the LCM of the denominators and adding up the numerators)
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