Cube Root of 216

We know that to find the volume of the cube, we have volume = side3, but to find the side of a cube we have to take the cube root of the volume.

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 \[4^{3} = 4\times 4\times \times 4 = 64\] or \[8^{3} = 8 \times 8 \times 8 = 512\] etc.

Thus, we can say that the cube root is the inverse operation of cubing a number. The cube root symbols is \[\sqrt[3]{}\] , it is the “radical” symbol (used for square roots) with a little three to mean cube root.

The cube root of 216 is a value which is obtained by multiplying that number  three times. It is expressed in the form of \[\sqrt[3]{216}\] . 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 \[\sqrt[3]{216} = \times\], then \[{\times}^{3}\] = 216 and we need to find here the value of \[\times\].

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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 ‘a’ itself. 

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

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

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

The symbol of the cube root is n3  or \[\sqrt[3]{n}\] 

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. 

Example: If the cube of a number 53 = 125

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

As 216 is a perfect cube, cube root of  \[\sqrt[3]{126}\] can be found in two  ways

Prime factorization method and Long Division method.

Calculation of Cube Root of 216

Let, ‘n’ be the value obtained from \[\sqrt[3]{126}\], then as per the definition of cubes, \[n \times n \times n = n^{3} = 216\]. Since 216 is a perfect cube, we will use here the prime factorisation method, to get the cube root easily. Here are the following steps for the same.

Prime Factorisation Method

Step 1: Find the prime factors of 216

\[216 = 2 \times 2 \times 2 \times 3 \times 3 \times 3\]

Step 2: 216 is a perfect cube. Therefore, group the factors of 216 in a pair of three and write in the form of cubes.

\[216 = {(2 \times 2 \times 2)} \times {(3 \times 3 \times 3)}\]

\[216 = 2^{3} \times {3}^{3}\]

Using the law of exponent, we get;

\[a^{m}b^{m} = (ab)^{m}\]

We get,

\[216 = 6^{3}\]

Step 3: Now, we will apply cube root on both the sides 

\[\sqrt[3]{216} = \sqrt[3]{6^{3}}\]

Hence,  \[\sqrt[3]{216} = 6 \]

Solved Examples

Example 1: Find the cube root of 512

Solution :

By Prime Factorisation method

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

\[512 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2\] 

Step 2: Form groups of three similar factors

\[512 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2\]

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

\[= 2^{3} \times 2^{3} \times 2^{3}\]

\[= 8^{3}\]

Therefore, \[\sqrt[3]{512} = 8 \]

Example 2: Find the cube root of 1728

Solution :

By Prime Factorisation method

Step 1: First we take the prime factors of a given number 1728 

\[= 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3\]

Step 2: Form groups of three similar factors

\[= 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3\]

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

= \[2^{3}\] \[\times\] \[2^{3}\] \[\times\] \[3^{3}\]

\[= 12^{3}\]

Therefore,  \[\sqrt[3]{1728} = 12\]

Quiz

1. Using prime factorization, find the value of   \[\sqrt[3]{1331} \]

2. Using long division method, find the value of  \[\sqrt[3]{729} \]

Simplify Algebraic Cube Root

To simplify algebraic cubic roots, the cubic radical must meet the following requirements:

• There should be no fractional value under the radical sign.

• Under the cube root symbol, there should be no ideal power factors.

• No exponent value should be bigger than the index value when using the cube root symbol.

• If the fraction appears under the radical, the fraction's denominator should not include any fractions.

When calculating the cube root of any integer, we will look for the components that appear in the set of three. For instance, the cube of 8 is 2. \[2 \times 2 \times 2\] is the factor of 8. Cube roots, unlike square roots, should not be concerned with the negative values under the radical sign. As a result, perfect cubes might have negative values. It is worth noting that perfect squares cannot have a negative value.

Use of Cube and Cube Roots

Several mathematical and physical operations employ cubes and cube roots. It's frequently used to find the solution to cubic equations. To be more precise, cube roots may be used to calculate the dimensions of a three-dimensional object with a given value. Cubes and cube roots are frequently employed in everyday math computations while studying topics such as exponents. Cube is also used to solve cubic calculations and get the dimensions of a cube given its volume.

Cube Root of a Negative Number

The prime factorization method is the best approach to get the cube root of any integer.

• Perform the prime factorization of the provided integer in the case of negative numbers as well.

• Divide the acquired factors into three groups, each containing the equal number of each component.

• To find the cube root, multiply the components in each group.

• It's simply that adding three negative numbers yields a negative result. It is indicated by the negative sign in conjunction with the cube root of a negative number.