Formula and Execution

Number

This is the value whose cube root or nth root is required. In this section, you will substitute the number to find its nth root.

Degree of Root

The degree of root is the number which, when multiplied by the same times as the root number, generates the original number. Like square root has root 2, and when it is multiplied 2 times, it generates the original number. Example: √x * √x = x. In this section, you substitute the number to which the root is required. For example: for square root, you will substitute 2; for cube root, you substitute 3, similarly, for fourth root, 4, and so on.

Root

This is the result section for all your answers!

Cube Root Calculator

Smart, handy, quick, precise, and free! Sounds like a complete package, right? Well, that is because it is. Our online cube root calculator computes all your cube root calculations instantly making your life a lot easier than you thought. And guess what? We are not just limited to cube roots, but to roots of any degree. So basically, it is an nth Root Calculator!

Just type in your values, and as the second you feed the calculator up, the second it generates results.

Besides, also check out full list of all the calculators that are all complete packages of all the good attributes just like this one, to make your math ventures a lot easier and fun for you.

How to Calculate Cube Root or nth Root by Using Calculator Beast

Substituting numbers in the right section is all it takes. Let us calculate the cube root of 5832. Just substitute in 5832 in the ‘Number’ section, 3 in the ‘Degree of Root’ section, and see the result in the ‘Root’ section! Here is the screenshot from our calculator for better illustrations.

Cube Root Calculator

To calculate the nth root of any number, we again don’t have to deal with climbing up the mountains. Follow the same steps as above, just instead of 3, type the number to which the root needs to be calculated. I have taken the example of 6th root for the number 1000. 

Another good news is that our cube root/nth root calculator works both ways. You can also substitute a number in the root section, and the degree of root to which the number is raised. And see your results in the number section. Brilliant, right?

Concept of Cube Root

Before you read further, I would appreciate it if you take a glance at the concepts described in our online square root calculator. It is not mandatory, but it just makes things easy, you know. And who doesn’t like easy?

Cubes

We all know what cubes are. The famous Rubik’s Cube is an example from our daily lives. So, when a number is raised to the power of 3, we say it is cubed. Why? Because it forms a cube in a 3-dimensional plane, as shown in the figure.

Cubes

We discussed in the square root calculator that squaring a number generates the area of a square. Similarly, cubing a number generates the area of a cube. The figure above shows 33 plotted in a 3-dimensional plane. 

Cube Root

Just as we saw that the square roots are the inverse of the squares, the cube roots, in the same way, are inverse of the cubes. We use the radical ‘√’ sign, the same way we use for the square roots. The difference is that we write a small 3 with the radical sign like ∛. Mathematically, we define cube roots as,

“If a number y can be expressed as y = x3, we say that y is the cube of and x and x is the cube root of y.”

∛y = x ó y = x3

Cube roots can also be written in the form of exponents as ∛y = y1/3.

Examples:

  1. Calculate the cube root of 9261.

Solution

The calculator at your left-hand side will make all your cube roots calculations a piece of cake! Substitute 9261 in the number section, 3 in the degree of root section, and I hope you have got 21 as a result in the root section.

This means when you multiply 18 thrice, you get 5832. So, we say that the cube root of 5832 in 18.

  1. Calculate the number whose cube root is 1.396.

Solution

So, by reading the problem, you already know that this time, we are going the other way round. Our calculator not only gives results for the cube roots, but it also generates cubes when roots are added to it. 

So, substitute 1.396 in the root section, 3 in the degree of root section, and what you got? 2.72, right? Well, that’s an approximation up to two decimal places of the value of the exponential constant ‘e. So, now we know that the cube root of e is 1.396.

Perfect Cubes

If we see the examples discussed above, the concept of perfect cubes will become easy to understand. In example 1, 9261 is the perfect cube of 21. And in the second example, 1.396 is not the perfect cube of the exponential constant.

Were you able to draw any conclusions from above? 

Well, if we calculate the cube root of any number, and the result is a whole number, then the original number is said to be the perfect cube of the result. And if the result on calculating the cube is not a whole number, then the original number is not the perfect cube of any number. 

Some Commonly Used Values of Cubes and Cube Roots

Calculators are a very handy tool when it comes to calculating the cubes and cube roots of bigger and irrational numbers. But we definitely would not recommend you to completely rely on these computation tools. 

One must learn the basic values of the cubes and cube roots

xx3x∛x
1111
2882
327273
464644
51251255
62162166
73433437
85125128
97297299
101000100010

How to Calculate Cube Roots by Prime Factorization?

Prime Factorization is something we all learn at elementary school levels. I assume you already know how to calculate prime factors; if not, our online GCF calculator is always there to teach you prime factorization.

We list down the prime factors of 1728,

1728 = 2.2.2.2.2.2.3.3.3 (here dot represents multiplication)

To calculate the cube root of 1728, we take cube root on both sides of the above equation

∛1728 = ∛(2.2.2.2.2.2.3.3.3)

              = ∛{(2.2.2)(2.2.2)(3.3.3)}

              = ∛{(2)3(2)3(3)3}

We already discussed that the cube roots could also be written in the form of exponents. So we write the above expression as,

∛1728 = [(2)3(2)3(3)3]1/3

                   = 2.2.3 = 12

So the cube root of 1728 is 12.

How to Estimate Cube Roots?

To estimate the cube root of any number, you first need to learn the cube root table written above. Once those values are at your fingertips, then you can easily leave your friends in awe by this trick. 

Notice the values in the table. Notice how the cubes for each number ends. Did you see anything observable? Well, the values of the cubes for the numbers 1,4,5,6,9, and 0, end with the same digits respectively. Like the cube root of 1 ends in 1, the cube root of 4 ends in 4, and the same goes for the rest of the numbers. 

While the cubes for 2 and 8, and 3 and 7, flip. Like cube root of 2 is 8, that is, it ends in 8, and the cube root of 8 is 512, i.e., it ends in 2. The same goes for 3 and 7. 

This pattern follows for all numbers. Once you have mastered this pattern, then as you see any number whose cube root is required, the first thing you do is to simply write the last number.

Let us suppose the number 46,656. The last digit is 6, so we know that our last number of the cube root would be 6. Now ignore the last three numbers, i.e., 656. Consider 46 and run through your brain the list you just memorized and pick the nearest cube. The nearest cube is 27, whose cube root is 3. Simply place this 3 before 6. So our cube root for 46,656 is 36. 

Let us dive into another example. If we have to find the cube root of 314,432, we see the last digit. It is 2, so we know that the cube root would end in 8. Now our last digit is known to us. Ignoring 432, we see the cube nearest to 314 from our list. It is 216, and its cube root is 6, so I guess you already know the cube root of 314,432. Yes, it is 68. No big a deal, right? Well, it might seem time-requiring in the beginning, but the more you practice, the more you ace at it. 

Graph of a Cube Root Function

The cube root function is odd i.e., it fulfills the condition f(-x) = -f(x). Being an odd function makes it symmetric about the origin. The graph of a cube root function also has a vertical tangent at x=0. 

Graph of a Cube Root Function

The Derivative of a Cube Root Function

The formula for carrying out basic differentiation is,

f(x) = xn ó f’(x) = nx(n-1)

If we consider the cube root function f(x) = ∛x, then by using the above formula, the derivative of the cube root function would be,

f’(x) = 1/3 (x)1/3-1

f’(x) = 1/3 (x)-2/3

f’(x) = 1/3 . 1/x2/3       (dot implies multiplication)

f’(x) = 1/(3∛x2)

Nth Root 

We studied square roots and cube roots. We also discussed why they are termed as square and cube. Now the math is not limited to the roots of these numbers. It, of course, goes on to infinity. 

We cannot name the root of every single number individually because no one would be able to remember infinite names, and it just makes things complex rather than easy. So, when we take the roots of numbers except for the square root and cube root, we simply say, we are taking the nth root of that number. In general notation, we also write n as the index of the root, like n√x. If we are to calculate the 4th root of any number, we say 4th root and write it as ∜x. The same goes on for the 5th, 6th, 7th, and so on for roots up to infinity. 

Our nth root calculator helps you solve your problems related to roots to any number. Just type in the number in the root section, nth root in the degree of root section, and get your results instantly in the root section. The screenshot below shows the 7th root of 356,897. 

nth root calculator

FAQs

How do I calculate the cube root by a scientific calculator?

If you are using a scientific calculator, then calculating either the cube root or the nth root is no big a deal. Find out the square root button. In most of the calculators, the square root button has cube root written above it, on the base of the calculator. This function works after pressing the shift button. So, press the shift button and now press the square root button. Type in the number whose cube root is required, press = sign, and bingo!

When it comes to calculating the nth root, find the key in your calculator that has a box in its index place. Since the calculator comes in different versions, so this button may exist at different places in different versions. Find out where it exists in your calculator. Press the button, now type the root in the index box, and the number under the radical by using the arrow keys to shift from index to under the radical sign. Press = sign and get your results for the nth root of any number.

How do I calculate the cube root on a regular calculator?

Well, the steps to calculate the cube root on a regular calculator are something you will have to memorize, but it is not something impossible, and also you can verify your answers during your exams if cube root is required to be calculated manually.

  • The first step is to type the number in the calculator.
  • The second step is to press the root sign twice.
  • Now press the multiply sign and again press root sign for four times.
  • Again press multiply sign and this time press the radical/root sign for eight times. Make sure you do not miscount.
  • This is the last time you have to press the multiply sign. Now press the radical sign twice.
  • Press = and get your results.

To memorize the steps, simply learn this list: type number, then press root sign two times, four times, eight times, and two times. During every comma, use multiplication operation. Hope this makes things easier to memorize.

Note: Slight variation in the answer may exist due to rounding off issues. Otherwise, the results are worth using. 

Do cube roots of negative numbers exists?

Yes, cube roots of negative numbers exist. And calculating them is easy in comparison to the square roots. In square roots, we studies about how complex numbers get involved when we calculate the square root of negative numbers. But in this case, things don’t get complex. 

Why? Because when we multiply a negative number three times, it gives us a negative result. So, what do you think would be the cube root of -216? -6, of course, because -6*-6*-6=-216. 

We have two results for square roots, positive and negative, what about cube roots and nth roots? 

The easiest way to know about the number of results existing while calculating the roots is simply to see the degree of root. If the degree is 2, results would be 2, if it is 3, we will have three results as its roots, if it’s 4, we have 4 roots, and if it’s n, we have n roots.

So you must be thinking, why do you get 1 result when you calculate the cube root? Well, the answer because that one result is the real result, the other two results are complex numbers, i.e., numbers in the form of a+ib, where i is the imaginary number equal to √-1. 

The imaginary numbers are usually not dealt with very often. So, most of the time, we ignore those roots until there is a need. And yes, although they are termed as imaginary, the numbers do exhibit great use in the real world, and that is why we have them. The mathematicians, physicists, and other scientists know where and how to use them.

Our calculator deals with real values only. The imaginary values take some time to calculate, so we won’t be going into the details of calculating them, but we will prove our point that they exist, by an example.

The cube roots of 27 are 3, -1.5+1.5(√3)i, and -1.5-1.5(√3)i. To prove these are the cube roots of 27, we simply take a cube of these numbers. If the result is 27, they are the cube roots.

  • For 3, (3)3 = 3*3*3 = 27
  • For -1.5+1.5(√3)i, we use the multiplication formula (a+b)3 = a3 + b3 + 3a2b + 3ab2 and we keep in mind that i2 = -1 and i3 = -i. 
    We consider a=-1.5 and b=1.5(√3)i
    (-1.5+1.5(√3)i)3 = (-1.5)3 + (1.5(√3)i)3 + 3(-1.5)2(1.5(√3)i) + 3(-1.5)( 1.5(√3)i)2
                                  = -3.375 – (81√3)i/8 + (81√3)i/8 -30.375
                                  = 27

You can carry out the calculation for the third value and see if you get 27 as the answer.