In math, the cube root of a number like 421 is a number that, when multiplied by itself two times, is equal to 421.
We would show this in mathematical form with the cube root symbol, which is similar to the radical symbol (√) used in a square root, but which also has the number 3 above the symbol (this number is called the index):
So, the √ is called the radical symbol, the little 3 above it is the index (which means multiply by itself two more times), and the number, 421, is called the radicand.
To explain the cube root a little more, the cube root of the number 421 is a number (which we call n) that when multiplied by itself twice is equal to 421:
∛421 is the same as 421⅓ and so if you are solving a problem in a spreadsheet like Excel, Numbers, or Google Sheets, you can use
=421^(1/3) to calculate the cube root.
For the purposes of this article, we'll use a scientific calculator to work out the cubic root by typing the number 421 and pressing the
[∛x] button, to give the following answer:
Is 421 a Perfect Cube?
When the cube root of a given number is a whole number, this is called a perfect cube. Perfect cubes are important for many mathematical functions and are used in everything from carpentry through to more advanced topics like physics and astronomy.
If we look at the number 421, we know that the cube root is 7.4948112258591, and since this is not a whole number, we also know that 421 is not a perfect cube.
If you want to learn more about perfect cube numbers we have a list of perfect cubes which covers the first 500 perfect cube numbers.
Is the Cube Root of 421 a Rational or Irrational Number?
Another common question you might find when working with the roots of a number like 421 is whether the given number is rational or irrational. Rational numbers can be written as a fraction and irrational numbers can't.
The quickest way to check if a number is rational or irrational is to determine if it is a perfect cube. If it is, then it's a rational number, but if it is not a perfect cube then it is an irrational number.
We already know that 421 is not a rational number then, because we know it is not a perfect cube.
Rounding the Cube Root of 421
Sometimes when you work with the cube root of 421 you might need to round the answer down to a specific number of decimal places:
10th: ∛421 = 7.5
100th: ∛421 = 7.49
1000th: ∛421 = 7.495
Simplifying the Cube Root of 421
It is possible to simplify the cube root of 421 if the radicand can be made smaller. When that is possible, it is called the cube root of 421 in its simplest radical form.
In this example, the cube root of 421 cannot be simplified down any further.
Practice Cube Roots Using Examples
If you want to continue learning about cubic roots, take a look at the random calculations in the sidebar to the right of this blog post.
We have listed a selection of completely random numbers that you can click through and follow the information on calculating the cubic root of that number to help you understand number roots.
Calculate Cube Square Root Problem
Enter your number in box A below and click "Calculate" to work out the cube root of the given number.