What Is the Value of Cube Root of 3 and How to Find It
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 Understanding the Cube Root of 3 in Detail
1. What is the cube root of 3?
The cube root of 3 is the number that, when multiplied by itself three times, equals 3, and its approximate value is 1.442. In radical form, it is written as ∛3. Since 3 is not a perfect cube, its cube root is an irrational number with a non-terminating, non-repeating decimal expansion.
2. How do you find the cube root of 3?
You can find the cube root of 3 using a calculator, estimation, or numerical methods.
- Using a calculator: Enter 3^(1/3) to get approximately 1.442.
- By estimation: Since 1³ = 1 and 2³ = 8, ∛3 lies between 1 and 2.
- Using long division or Newton’s method for more precise manual calculation.
3. Is the cube root of 3 a rational number?
The cube root of 3 is irrational because it cannot be expressed as a ratio of two integers. Its decimal form, approximately 1.442249..., is non-terminating and non-repeating. Since 3 is not a perfect cube, its cube root cannot be a whole number or rational fraction.
4. What is the value of cube root of 3 up to three decimal places?
The value of the cube root of 3 up to three decimal places is 1.442. The more accurate value is approximately 1.442249, but rounding to three decimal places gives 1.442.
5. How do you write the cube root of 3 in exponential form?
The cube root of 3 in exponential form is written as 3^(1/3). This follows the exponent rule:
- a^(1/n) = ⁿ√a
6. What is the cube of the cube root of 3?
The cube of the cube root of 3 is exactly 3. Mathematically:
- (∛3)³ = 3
- Or (3^(1/3))³ = 3
7. What is the difference between √3 and ∛3?
The difference between √3 and ∛3 is that √3 is the square root while ∛3 is the cube root of 3.
- √3 ≈ 1.732 (number multiplied by itself twice).
- ∛3 ≈ 1.442 (number multiplied by itself three times).
8. Is 3 a perfect cube?
No, 3 is not a perfect cube because there is no integer whose cube equals 3. Perfect cubes include numbers like 1 (1³), 8 (2³), and 27 (3³). Since 3 lies between 1 and 8, its cube root is not a whole number.
9. Can you simplify the cube root of 3?
The cube root of 3 cannot be simplified further because 3 has no perfect cube factors other than 1. Since 3 is a prime number, ∛3 is already in its simplest radical form.
10. What are some properties of the cube root of 3?
Some important properties of the cube root of 3 follow the general cube root rules.
- (∛3)³ = 3
- ∛(3 × a) = ∛3 × ∛a
- ∛(3/a) = ∛3 / ∛a
- It is an irrational number.
