How to Calculate the Cube Root of 3: Methods and Examples
When we say the cubed root of 3, then it would mean 27. This is calculated as (3x3x3). When a number is cubed, it implies that it is multiplied itself by 3 times. So 3 cubes is 27. For example, If you have 7 to the power of 3 (or cubed, as 3 implies cubed in math) then to solve, I would say 7 multiplied by itself 3 times. 7x7x7, this comes out to be 343(7x7 is 49, and 49x7 is 343).
How to Determine the Cube Root of 3?
Before we proceed to learn finding the cube root of number 3, we have to first acquaint ourselves in knowing the cubes of the numbers stated below.
Cube Root Lists
Below is a tabular form for the cube root of numbers from 1 to 15.
Cube Root of 3 Explained
The cube root of any number n will be a number x, such as x³ = n. Thus, in order to identify the cube root of three, we are required to determine a number, which when multiplied three times with the number itself, provides the number 3, such as x³ = 3 or x = \[\sqrt[3]{3}\]. Hence, we will here require finding the value of x.
The value of the cube root of 3, \[\sqrt[3]{3}\], is equivalent to 1.44224957031. Seeing that 3 do not make a perfect cube, hence it is a bit difficult to determine its cube root. However, for perfect cubes like 8, 27, 64, 125 729 etc., the cube root of such numbers are whole numbers, since, 3³ = 3×3×3 = 27 and 5³ = 5×5×5 = 125. Therefore, the cube root of 125 is 5 and of 27 is 3.
Thus, for 3 in cubed, then you would multiply it 3 times. 3³ = 3x3x3; 3x3 is 9, and 9x3 is 27. Hence, the 3 cubed is 27.
Cube Root Symbol
The symbol of cube root is as ‘\[\sqrt[3]{}\]’
Cube root can also be determined by the approximation method.
Step-By-Step Process to Find Cube Root Easily
Now, let us find out the value of \[\sqrt[3]{3}\], in a detailed step by step manner.
Let us suppose that the cube root of 3 is equivalent to x.
Then, x = \[\sqrt[3]{3}\]
As we are already aware,
1³ = 1 and 2³ = 8
Thus, x lies in between 1 and 8. But, it lies near to the number 1 than 8, if we observe in a number line. Therefore, we can presuppose a value; say 1.4, which could be an approximation to the cube root of 3.
Now, in order to identify the value of \[\sqrt[3]{3}\], we are required to divide 3 by an approx value.
Dividing 3 by 1.4, we get
3/1.4 = 2.1428
Again, we need to divide this value by 1.4.
On dividing, we get,
2.1428/1.4 = 1.53
Now, taking the average of 1.4, 1.4 and 1.53 in order to obtain the value of \[\sqrt[3]{3}\].
(1.4+1.4+1.53)/3 = 1.44
Thus, the actual value of \[\sqrt[3]{3}\] = 1.442249.
Therefore, 1.44 is approximately equivalent to 1.442249.
FAQs on Cube Root of 3 Explained
1. What is the value of the cube root of 3?
The cube root of 3, denoted as ∛3, is the number that results in 3 when multiplied by itself three times. Since 3 is not a perfect cube, its root is an irrational number. The value of the cube root of 3 is approximately 1.4422. In radical form, it is written as ∛3, and in exponential form, it is expressed as 3¹ᐟ³.
2. How can you estimate the cube root of 3 without a calculator?
To estimate the cube root of 3, you can identify the perfect cubes it lies between. We know that:
- 1³ = 1
- 2³ = 8
Since 3 is between 1 and 8, its cube root must be between 1 and 2. By testing values, we find that 1.4³ is approximately 2.744 and 1.5³ is 3.375. This shows the value of ∛3 is much closer to 1.4. This estimation method is a fundamental technique for understanding the magnitude of roots for non-perfect cubes.
3. Why is the cube root of 3 considered an irrational number?
The cube root of 3 is an irrational number because it cannot be expressed as a simple fraction (p/q, where p and q are integers). Its decimal representation is both non-terminating and non-repeating, meaning the digits continue infinitely without a recurring pattern. This property is characteristic of the roots of most non-perfect cube numbers.
4. What is the key difference between the 'cube of 3' and the 'cube root of 3'?
The 'cube of 3' and the 'cube root of 3' are inverse mathematical operations with distinct meanings and results.
- Cube of 3 (3³): This involves multiplying 3 by itself three times: 3 × 3 × 3 = 27.
- Cube root of 3 (∛3): This is the number that, when cubed, equals 3. Its value is approximately 1.442.
5. How does the cube root of 3 (∛3) compare to the square root of 3 (√3)?
Both ∛3 and √3 are irrational numbers, but they originate from different operations and have different values.
- The Square Root of 3 (√3) is the number that, when multiplied by itself, gives 3. Its value is approximately 1.732.
- The Cube Root of 3 (∛3) is the number that, when multiplied by itself three times, gives 3. Its value is approximately 1.442.
6. Is the cube root of 3 a real number?
Yes, the cube root of 3 is a real number. The set of real numbers is composed of all rational and irrational numbers. Since ∛3 is an irrational number, it is included within the definition of real numbers. Every positive number has exactly one real cube root.
