Mathematics Tutor and Cube Root

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What is a Cube Root of a Number?

When you first begin learning about cube root, the math can seem a bit confusing. But it’s not as hard as it looks—it’s just a lot of numbers! Here are some ways to start thinking about cube roots, which should help clear up what they are.

  • The cube root of a number is defined as the number that when multiplied 3 times by itself gives us our original number.
  • For example, if we were to multiply 2 x 2 x 2 (or 2 cubed), we would get our original number: 8! So, 2 is the cube root of 8 because 2 x 2 x 2 = 8.
  • There is also a symbol for cube root that looks like this: ∛. This symbol consists of the radical sign (√) with a small three in the top right corner called an index. When you see something with an index, you know that it refers to how many times you will use the base as a factor—in this case, 3. So ∛8 means “the cube root of 8.”

Examples of Cube Root

  • Cube Root of 8: This example I solved by taking the cube root of 8. Cubing is the equivalent of taking the cube root of 8, just the inverse. Therefore, it is the number that needs to be multiplied three times to create a cube.

Here’s how you can solve this problem:


Cube root (8)=23=2

The answer is 2.

Perfect Cube Root

The cube root of a number [math]x[/math] is a number [math]a[/math] such that [math]a^3 = x.[/math]. If you find that the cube root is negative then it means that there is no real answer, since cubing a negative number gives a positive result. Note: Finding the cube root is the inverse of cubing, however cubing is not the inverse of finding the cube root.
If we were to take the cube root of 27, this would be 3 like 3 times 3 times 3 = 27. Whereas if we were to find -27 cubed this would be -27 (as (-27) x (-27) x (-27) = –19683).

Formula for Cube Root

When you are trying to figure out the cube root of a number, you are trying to find one value that when multiplied three times by itself will give you that number. A simple example would be if we start with 8, because 2 × 2 × 2 = 8, then the cube root of 8 is 2.
Mathematically, a cube root of x is a number whose third component is the same as the original number. Despite their zeroes, real numbers (except for zero) have one real cube root and a pair of complex conjugate cube roots, and nonzero complex numbers have three distinct complex cube roots.
The formula for the volume of any cuboid is length × width × height or lwh. Since volume must be an integer (a whole number), it follows that lw or wh must be perfect cubes themselves; only, in this case, will their product also be so. In other words, if c is the largest integer such that c3 divides both l and w, then c3h must be a volume!

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