How to Calculate the Cube Root of 216 Easily
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
Using prime factorization, find the value of \[\sqrt[3]{1331} \]
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.
FAQs on Cube Root of 216 Explained
1. What is the cube root of 216?
The cube root of 216 is 6. This is because when 6 is multiplied by itself three times (6 × 6 × 6), the result is exactly 216. A number like 216, which has a whole number as its cube root, is known as a perfect cube.
2. How do you find the cube root of 216 using the prime factorisation method?
To find the cube root of 216 using the prime factorisation method, you must break down 216 into its prime factors and group them into triplets. The steps are as follows:
- First, find the prime factors of 216: 216 = 2 × 2 × 2 × 3 × 3 × 3.
- Next, group the identical factors into sets of three: (2 × 2 × 2) and (3 × 3 × 3).
- This can be written in exponential form as 2³ × 3³.
- Finally, take one factor from each group and multiply them: 2 × 3 = 6.
Therefore, the cube root of 216 is 6.
3. How can you prove that 216 is a perfect cube?
You can prove that 216 is a perfect cube by checking if all its prime factors can be perfectly grouped into sets of three. The prime factorisation of 216 is 2 × 2 × 2 × 3 × 3 × 3. Since these factors form complete triplets of (2³) and (3³) with no factors left over, it confirms that 216 is a perfect cube.
4. What is the difference between the square root and the cube root of 216?
The primary difference between the square root and cube root of 216 lies in the number of identical factors required.
- The cube root of 216 is 6, as it is the number that, when multiplied by itself three times (6 × 6 × 6), equals 216.
- The square root of 216 (√216) is approximately 14.69. It is not a whole number because 216 is not a perfect square. The square root seeks a number that, when multiplied by itself just twice, equals 216.
5. Why is the prime factorisation method preferred for finding the cube root of perfect cubes like 216?
The prime factorisation method is preferred for perfect cubes because it provides a definitive and visual proof. It breaks the number down into its fundamental prime factors. If all these factors can be arranged into identical triplets, it confirms the number is a perfect cube and easily reveals the root. Other methods, like estimation, are more practical for non-perfect cubes where an exact whole number answer doesn't exist.
6. Can a number have a negative cube root? If so, what is the cube root of -216?
Yes, a number can have a negative cube root. The cube root of a negative number will always be negative. The cube root of -216 is -6. This is because multiplying a negative number three times results in a negative product: (-6) × (-6) × (-6) = 36 × (-6) = -216. This is a key difference from square roots, as you cannot find a real square root for a negative number.
7. How does the unit digit of 216 help in identifying its cube root?
The properties of cubes show that the unit digit of a perfect cube is determined by the unit digit of its root. The number 216 has a unit digit of 6. The only single digit that results in a cube ending in 6 is 6 itself (since 6³ = 216). This property serves as a valuable shortcut for estimation, instantly telling you that the cube root of 216 must be a number ending in 6.
