Formula and Execution

Number

In this section, you will plug in the base of the number. Your number is of the form a = bx. This section needs you to substitute b.

To the Power of (x)

This section requires you to feed it with the power, i.e. the number which is in place of x in the form a = bx.

Result (a)

You get to see your results in this section, which is the value of a.

Exponent Calculator

Our straightforward online exponent calculator helps you get your required results in just a heartbeat! Any base raised to any power can be easily computed by plugging in the desired values in the required columns. 

Not just this, our online exponent tool works like a magic spell that gives you results not only for the number raised to any power but for any required value. The exponent form is mathematically expressed as a = bx. You can get your hands on any value by plugging in any two values in the right section. 

If you want a, plug in values for and x. If you want b, plug in values forand x. Similarly, if you want x, plug in values of and b

How to Calculate Exponent using Calculator Beast 

Well, to carry out this task, the first step is to visit this webpage for sure. The next thing is to plug in the right values in the right section and ta-da! A screenshot will to illustrate you that the calculator not only works for positive numbers, but for negative numbers, too.

Exponent Calculator

Concept of Exponents

To create a better understanding of the exponents, we consider a simple example. Consider that you have to write 5 multiplied by 5. This is how you will write it:

5 x 5

Now consider you have to write 5 multiplied by 5 multiplied by 5. So this is how we go:

5 x 5 x 5

Imagine you have to write this thing where 5 is supposed to be multiplied nine times.

5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5

It is just getting heftier, ain’t it? Imagine carrying multiplication where 5 is multiplied by itself for 50 times. Well, I ain’t writing it. 

This is exactly what the mathematicians thought when they had to deal with writing such repetitive multiplications. So, what they did was to introduce a writing way to ease the process. This writing way is termed as exponentiation or just exponents. 

How did they make it easier? Well, when 5 was multiplied by 5, they wrote it as 52, when 5 was multiplied by 5, and it was again multiplied by 5, they wrote this as 53, and similarly in the above example, when 5 is multiplied 9 times to itself, they used this form and represented the number like 59. The number which is written as a superscript is the number of times, the number is multiplied to itself. 

Generally, we write this as,

a = bx

Where b is known as the base, x is known as the index or power or exponent, and it the number of times the base is multiplied to itself, and a is the result that we get by carrying out this operation. Like in 53 = 125, 5 is the base, 3 is the power or exponent, and 125 is the result.

Sometimes, when we are writing in our phones, our keyboards won’t allow us to write in the form of power. So, we use the ‘^’ symbol to write an exponent form. Like 53 can also be written as 5^3. If the powers are in fraction form, make sure to use brackets to avoid misunderstandings. It is easy to understand 52/3, but when you write 5^2/3, you are not sure if the whole fraction is in power or just 2 is in power, and then we are supposed to divide the quantity by 3. So, to avoid such misunderstandings, the use of brackets is very useful. Notice the difference in the two forms: 5^(2/3) and 5^(2)/3. Remember, these are different from one another! 

We have designed this calculator to help you ease your problems that involve bigger numbers, decimals, or negative numbers. The smaller and simpler ones can be easily computed in your mind, but visit us when you think you are stuck in your difficult ventures, and get precise results!

Interesting Exponents!

We saw what exponents mean, and if any number is raised to some power, that number is multiplied by itself that many times. Like, 62 = 6 x 6 and 84 = 8 x 8 x 8 x 8.

What if the number is raised to the power of 1? Well, that is simple to answer as well. It is simply the number itself. Like 41 = 4. No big a deal, right?

So, what gets interesting exactly? When the number is raised to the power of o, that case gets interesting. How? Let us dive into it.

Consider the following set of exponents of 6:

65 = 6 x 6 x 6 x 6 x 6
64 = 6 x 6 x 6 x 6
63 = 6 x 6 x 6
62 = 6 x 6
61 = 6

Did you notice any pattern? Every time we reduce the number in the exponent, we are dividing the number by itself. Like five times 6 when divided by 6 gives four times 6 and the exponent becomes 4, four times 6 when divided by 6 gives three times six and the exponent becomes 3, and so on. So, what when one time 6 is divided by 6? That is like dividing the number by itself. So, the answer is always 1. If I make a statement like previous ones, it would be like; when one time 6 is divided by 6, we are left with 0 six and zero exponents. And we already know that 6 divided by 6 is 1.

So, we write the explanation as,

60 = 1

We give a generalized statement of the above discussion and say,

(Any non-zero number)0 = 1

Why non-zero? Why don’t we say that 00 = 1?

Well, there is a debate here between mathematicians. Some say it is equal to 1, and some say it equal to 0, so in general, everyone says that 00 is indeterminate. 

But why exactly is the debate on it? Why don’t we have a conclusion? For this, consider the following table.

111
0.90.90.909
0.80.80.836
0.70.70.779
0.60.60.736
0.50.50.707
0.40.40.693
0.30.30.696
0.2o.20.725
0.10.10.794
0.010.010.954
0.0010.0010.993

Did you notice anything? As we reduce the numbers and their powers, we are approaching near to zero. Then all of a sudden, a drastic change occurs in the pattern, and it flips, i.e., the numbers are again approaching towards 1. Notice this after the value of 0.40.4. 

So, this mystery remains the topic of debate in people, and they cannot come up with a solution. Therefore, they say, 00 is indeterminate.

What is 0 raised to any other number? Well, it is zero. 

03 = 0 x 0 x 0 = 0
02000 = 0

Since zero multiplied by itself as many times gives o. So, we generalize this as,

0(any positive number) = 0

Why positive, and why not negative numbers? That we will discuss while discussing negative exponents.

Another interesting exponent is anything raised to the power of 1. 

16 = 1 x 1 x 1 x 1 x 1 x 1 = 1

No matter how many times we multiply 1 by itself, the answer is always 1. So, in general,

1(any number) = 1

To summarize all the interesting exponents we learned above, we say:

  1. (Any non-zero number)0 = 1
  2. 00 = indeterminate
  3. 0(any positive number) = 0
  4. 1(any number) = 1

Laws of Exponent

Laws of exponents, also known as the rules of exponents, are some laws that help make our algebraic calculations easier.

  1. xm xn = xm+n
    This can be understood by an example. Consider the case, 32 x 34 = (3 x 3) (3 x 3 x 3 x 3) 
    We notice that the total number of times 3 is multiplied by itself is 6, so we write the above expression as, 32 x 34 = (3 x 3) (3 x 3 x 3 x 3) = 36 = 32+4
  2. xm/xn = xm-n
    To elaborate this, consider 34/32 = (3 x 3 x 3 x 3)/(3 x 3)
    Here, (3 x 3) in the numerator gets cancelled by (3 x 3) in the denominator, so, we are left with, 34/32 = (3 x 3 x 3 x 3)/(3 x 3) = 3 x 3 = 32 = 34-2
  3. (xm)n = xmn
    Understand this by the example, (32)4 = (3 x 3)4
    The above expression shows that (3 x 3) is multiplied by itself 4 times, which is written as, {(3 x 3)(3 x 3)(3 x 3)(3 x 3)}. This makes a total of eight 3s multiplied by itself. So, (32)4 = (3 x 3)4 = {(3 x 3)(3 x 3)(3 x 3)(3 x 3)} = 38 = 32x4

The above laws are valid only when we have the same bases. When we have the same index or exponent, then the laws stated are,

  1. (xn)(yn) = (xy)n
    We see this by the following example, (34)(24) = (3 x 3 x 3 x 3)(2 x 2 x 2 x 2) = (3 x 2)(3 x 2)(3 x 2)(3 x 2) = (3 x 2)4
  2. xn/yn = (x/y)n
    The example we consider here is, 32/42 = (3 x 3)/(4 x 4) = (3/4)(3/4) = (3/4)2

The last thing to know is to solve exponents of exponents, i.e., numbers of the form, x(a^b). To solve such numbers, we first calculate a^b. When the answer to this is known, then the next step is to raise the base to the obtained answer and calculate further.

Example: 2(4^2) = 216 = 65,536

Negative Exponents

To understand the negative exponents, we take the example of a number 4-2. We multiply and divide this number by 42, because that way, we can reach a conclusion easily without putting any mathematical impact on the number.

  • 4-2 x 42/42
  • (4-2 x 42) /42
  • (4-2+2) / 42 (by the law of exponent)
  • 40/42
  • 1/42 (anything raised to the power zero is equal to 1)

S0, 4-2 = 1/42

So, whenever we have negative power, we can reciprocate it to make it positive. We can also go vice versa. Like 42 can be written as 1/4-2.

To generalize, we say.

x-n = 1/xn or xn = 1/x-n

Now that the concept of negative exponents is clear, we see what happens when we raise a negative power to o.

0-n = 1/0n = 1/0 (here n is any number other than 0; we have already discussed the case of o0)

Since any number divided by zero is undefined, so 1/0 is undefined. Hence, whenever we see 0 raised to a negative exponent, we say it is undefined.

0(any negative number) = undefined

Our exponent calculator calculates both the positive and negative exponents unless the case is indeterminate or undefined. 

For insights that are similar to the topic of exponents, check out our square root calculator and cube root calculator that fall in the category of fractional exponents, and also don’t forget to peep into our logarithm calculator which is the inverse of exponents.

Graphs of Power Function

The graph of the power function varies with respect to the numbers we have in power. If the numbers in power are even, then the function is even, and the graph is symmetric with respect to y-axis. And if the numbers in the powers are odd, then the graph is odd, and it is symmetric with respect to the origin.

Given below are two separate illustrations for the graphs with even and odd powers, respectively.

    

Graphs of Power Function

           

Graphs of Power Function

If you notice the graph of the function with even powers, the bigger the value of n, the closer it gets to the axes. The graph of x6 is closer to the x and y axes as compare to the graph of x4 and x2. The same goes for x4 when compared to x2. 

When we take a look at the graph of the function with odd powers, we see an exact similar pattern as we saw in even function. The bigger the value, the closer it gets to the axes. 

FAQs

Are exponent and power the same?

Exponent, power, or index; all these are the different names of the same things. So, yes, these are the same. Exponents are also known as indices, but most of the time, the words commonly used are exponents and powers. 

Are power functions linear?

A linear function is something which is a one-degree function, i.e., it is raised to the power of one. When we plot such functions, the graph is a straight line. Therefore, we call it linear. So, the power function that has 1 as the maximum power raised to it are all linear. In the section above, where we have discussed graphs, you can see in the odd graph, the graph of x. It is linear.

Are exponential functions continuous?

Exponential functions are defined as a = bx. This function is defined for real values, that is, whatever value of x you plug in, you always get a defined result. If you substitute zero, you get 1, if you substitute values greater than 1, your results will be monotonically increasing, if you take x to be less than 0, the values will be monotonically decreasing, but you will always get a defined value. The functions that are defined for all real values are continuous, so, yes, exponential functions are continuous

Are exponents and logarithms related?

The inverse of addition is subtraction; the inverse of multiplication is division. Similarly, the inverse of exponents is logarithms. Exponent is knowing how to use power operation and getting a result. While logarithm is to know the power that yields the results. To learn more about logarithms, visit our logarithm calculator.

Can exponents be fractions?

The answer is, why can’t they be? Yes, exponents can be fractions, and they are known as fraction exponents. We have a fraction exponent calculator, too. Check that out and know more about fraction exponents and have fun by playing with the calculator.

What is the biggest number?

Since numbers have no limit and they can go up to infinity, yet, a mathematician came up with the idea of the largest number and named it Googol. It is 1 raised to the power of 100, i.e., 1100. To know more about it, visit the antilog calculator.