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Cube Roots

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 superscriptÂ³. Examples are 4Â³ = 4*4*4 = 64 or 8Â³ = 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 symbols is âˆ›.

The cube root of 4 is a value which is obtained by multiplying that numberÂ three times It is expressed in the form of â€˜âˆ›4â€™. The meaning of cube root is basically the root of a number which is generated by taking the cube of another number. Hence, if the value of âˆ›4 = x, then x3 = 4 and we need to find here the value of x.

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What is Cube Root?

The cube root of a number a is that number which when multiplied by itself three times gives the number â€˜aâ€™ itself.Â

For Example, 23 =8, or the cube root of 8 is 2

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

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

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

Thus, the cube root of 8 is represented as \[\sqrt[3]{8}\] = 2 and that of 27 is represented as \[\sqrt[3]{27}\] = 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 is the inverse operation of cubing a number.Â

Example: If the cube of a number 53 = 125

Then cube root of \[\sqrt[3]{125}\] = 5

Cube root of largest number can be found in twoÂ ways

Prime factorization method and Long Division method.

Properties of Cube Root

Cube root of all the odd numbers is an odd number. For example: \[\sqrt[3]{27}\] = 3, \[\sqrt[3]{125}\] = 5.

Cube root of all the even natural numbers is even. For example: \[\sqrt[3]{8}\] = 2, \[\sqrt[3]{64}\] = 4.

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

Calculation of Cube Root of 4

If n is a perfect cube for any integer m i.e., n = mÂ³, then m is called the cube root of n and it is denoted by m = âˆ›n.

Since 4 is not a perfect cube, hence we cannot use here prime factorisation method or estimation method to find the cubic root of 4. Therefore, we are going to use another method here which is called the Newton Raphson method. Here are the following steps for the same.

Step 1: Let us assume a number, say n, which is equal to the cube root of 4.

Here let us consider n=2 equal to the cube root of 4.

Step 2: Now, divide 4 by n and then divide its quotient again by n.

4/2 = 2 (first quotient)

and 2/2 = 1 (second quotient)

Step 3: Take the average of n and the two quotients obtained by division method. This value will be almost nearer to the value of \[\sqrt[3]{4}\].

Hence, we get here three numbers to generate the average.

The number 2 (assumed number), 2 (first quotient) and 1 (second quotient).

(2+2+1)/3Â

= 5/3Â

= 1.67

This value is almost near to the actual value of \[\sqrt[3]{4}\] , i.e. (1.5874)

Step 4: For more accurate value, we will repeat the method.

Now, we will repeat the method here by assuming n=1.6.

Thus,

4/1.6 = 2.5

and 2.5/1.6 = 1.5625

Taking the average of 1.6, 2.5 and 1.5625, we get;

(1.6+2.5+1.5625)/3Â

= 5.6625/3Â

= 1.8875.

Therefore, we get the value of the cube root of 4 equal to 1.8875, which is an almost accurate number.

Solved Examples

Example 1: Find the cube root of 2744

Solution :

By Prime Factorisation method

Step 1: First we take the prime factors of a given number

2744 = 2 x 7 x 2 x 2 x 7 x 7

Step 2: Form groups of three similar factors

= 2 x 2 x 2 x 7 x 7 x 7

Step 3: Take out one factor from each group and multiply.

= 23 x 73

= 143

Therefore, \[\sqrt[3]{2744}\] = 14

Example 2: Find the cube root of 1728 by long division method

Now,

\[\sqrt[3]{1728}\] = \[\sqrt[3]{2 \times 2 \times 2Â \times 2Â \times 2 \times 2Â \times 3Â \times 3Â \times 3}\]

= 2 x 2 x 3

= 12

Quiz Time

What is the cube root of 729?

What is the cube root of 214?

FAQ (Frequently Asked Questions)

1. What is the Difference Between the Square Root and Cube Root?

Answer:

The square root of a number x is that number which when multiplied by itself gives the number x itself.For example, the square root of 4 is 2.

The cube root of a number is a number which when multiplied three times gives you the number. For example, the cube root of 216 is 6.

The cube root of a number is the length of a cube whose volume is a

^{3}units. Whereas the square root of a number is the length of a square whose area is a^{2}units.The square root is an inverse operation of a squaring a number, while cube root is an inverse operation of cubing a number.

The square root is dented by âˆša, while cube root is denoted by 3âˆša.

2. What is Square Root?

Answer:

The square root of a number x is that number which when multiplied by itself gives the number x itself. The number x is a perfect square.

For Example, 2^{2} =4, or the square root of 4 is 2

Â Â Â Â 3^{2} =9, or the square root of 9 is 3

Â Â 4^{2} = 16, or the square root of 16 is 4

The symbol of the square root is âˆšÂ

Therefore, the square root of 4 is represented as âˆš 4 = 2.

And the square root of 9 is represented as âˆš 9 = 3 and so on.