Cube Root

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The process of cubing is similar to squaring, only that the number is multiplied three times instead of two times as in squaring. The exponent used for cubes is 3, which is also denoted by the superscript³. Examples are 4³ = 4*4*4 = 64 or 8³ = 8*8*8 = 512 etc.

To find the volume of the cube, we have volume = side3, but if we want to find the side of a cube we have to take the cube root of the volume. Thus, we can say that the cube root is the inverse operation of cubing a number. The cube root symbol is \[\sqrt[3]{}\].

Let’s suppose we need to find the value of cube root of 2 is a value that is obtained by multiplying that number three times It is expressed in the form of ‘\[\sqrt[3]{2}\]’. The meaning of cube root is basically the root of a number that is generated by taking the cube of another number. Hence, if the value of \[\sqrt[3]{2}\] = x, then x3 =2 and we need to find here the value of x.

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We can define the cube root of a number as a special value that, which when used in a multiplication exactly three times, gives us that number.

For example, 3 × 3 × 3 equals 27, so the cube root of 27 is 3.


The Cube Root Symbol

The special symbol given below signifies the "cube root", it is known to be the "radical" symbol (the symbol can be used for square roots) and with a little three to mean cube root.

\[\sqrt[3]{}\]

You can use it like this, the cube root of 27 is : \[\sqrt[3]{27}\] = 3(we say "the cube root of 27 equals to 3")


You Can Also Cube Negative Numbers

Have a look at this:

When we cube +5 we generally get +125:  +5 × +5 × +5 = +125

When we cube −5 we get the number −125:  −5 × −5 × −5 = −125

So the cube root of the number −125 is equal to −5


Cube Roots (For Integer Results 1 through 10)

  • Cube root of 1 is 1

  • Cube root of 8 is 2

  • Cube root of 27 is 3

  • Cube root of 64 is 4

  • Cube root of 125 is 5

  • Cube root of 216 is 6

  • Cube root of 343 is 7

  • Cube root of 512 is 8

  • Cube root of 729 is 9

  • Cube root of 1000 is 10


What is the Meaning of Cube Root?

The cube root of a number a is that number which when multiplied by itself three times gives the number ‘a’ itself. 

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Let’s see for example, 

23 =8, or the cube root of the number 8 is 2

33 = 27, or the cube root of the number 27 is 3

43 = 64, or the cube root of 64 is 4

53 = 125, or the cube root of 125 is 5

The symbol of the cube root is a3  or \[\sqrt[3]{a}\]

Thus, the cube root of 125 is represented as \[\sqrt[3]{125}\] = 5 and that of 27 can be represented as \[\sqrt[3]{27}\] equals 3 and so on.

We know that the cube of any number is found by multiplying that number three times. And the cube root of a number can be defined as the inverse operation of cubing a number. 


For Example:

If the cube of a number 63 = 216

Then the cube root of 216 is equal to 6.

Cube root of any largest number can be easily found in four ways:

  1. Prime factorization Method 

  2. Long Division Method

  3. Using Logarithms

  4. Bisection Method


A Few Properties of Cube Root

  • The cube root of all the odd numbers is an odd number. 

For example ∛125 = 5, ∛27 = 3.

  • Cube root of all the even natural numbers is even. For example: ∛8 = 2,∛64 = 4.

  • The cube root of a negative integer always results in negative.

Let’s Know Cube Roots of Some Numbers

Cube root of 16

The cube root of 16 is 2.1598

Cube root of 5

The cube root of 5 is 1.7099

Cube root of 6

The cube root of 6 is 1.1871

Cube root of 10

The cube root of 10 is 2.1544

Cube root of 12

The cube root of 12 is 2.2894

Cube root of 7

The cube root of 7 is 1.9129

Cube root of 0

The cube root of 0 is 0

Cube root of 20

The cube root of 20 is 2.7144


What are Perfect Cubes?

The example that we just saw also happens to be an example of a perfect cube. A perfect cube can be defined as a cube of a whole number. 27 is a perfect cube because to get the number 27, we need to cube the number 3. Think back to the cube. It's a perfect cube because all the building blocks are whole pieces. To find a perfect cube, we take any whole number and cube it, meaning we multiply it by itself three times. Knowing the perfect cubes will help us to find cube roots easily. If we started with the 1 and found perfect cubes for our numbers up to 10, we would get this list:


1

2

3

4

5

6

7

8

9

10

Perfect Cube

1

8

27

64

125

216

343

512

729

1000


Questions to be Solved

Question 1) What is the Cube Root of 30?

Answer

Well, 3 × 3 × 3 = 27 and 4 × 4 × 4 = 64, so we can guess the answer is between 3 and 4.

  • Let's try the following 3.5: 3.5 × 3.5 × 3.5 = 42.875

  • Let's try  the following 3.2: 3.2 × 3.2 × 3.2 = 32.768

  • Let's try the following 3.1: 3.1 × 3.1 × 3.1 = 29.791

Now we are getting closer, but slowly at this point, we can use a calculator and it says:

3.1072325059538588668776624275224...... but the digits just go on and on, without any kind of pattern. So even the calculator's answer can be known only as an approximation !


Question 2) What is the Cube Root of 1728?

Answer

The factors of 1728 are given as,

1728 = 12 × 12 × 12

∛1728 = ∛(12 × 12 × 12)

∛1728 = 12

FAQ (Frequently Asked Questions)

1. How Do I Find the Cube of a Number?

In arithmetic and algebra, the cube of any number suppose n is known to be its third power of that number: the result of the number multiplied by itself twice: n3 = n × n × n. You can also define it as  the number multiplied by its square: n3 = n × n2.

2. What is a Cube Number Example?

Cube Number can be defined as the result of using a whole number in a multiplication three times. Example: 3 × 3 × 3 = 27, so we can say that the number 27 is a cube number.

3. What is the Cube of 4?

The cube of 4 is 64

Finding a cube of any number means multiplying the digit with itself thrice. The formula for finding cube ( n3=n×n×n) and as per your question -: cube of 4 is 4×4×4=64.

4. What Does 3 Cubed Look Like?

When you multiply a whole number (not a fraction) by itself, and then by itself again the result is a cube number. For example, 3 x 3 x 3  is equal to 27. We can write the cube of 3 as 33. This means three multiplied by itself three times gives us 27.