What is the Cube Root of 2 Definition Value and How to Find It
The process of cubing is similar to squaring, the only difference is that the number is multiplied three times in the case of cubing and in squaring the number is multiplied two times. The exponent used for cubes is 3, which is also denoted by the superscript ³. For example, cube of a number can be expressed as 43 = 4*4*4 = 64 or 8³ = 8*8*8 = 512.
We can say that the cube root is the inverse operation of cubing a number. The cube root symbol is ∛.
In Mathematics, the cube root of a number (a) is the number (b) which satisfies the equation written below.
a3 = b
It can also be written in the form of
a = ∛b.
In this article, we will discuss about the meaning of cube root, its properties, the value of cube root of 2, and how to find the cube root of 2. We have also provided some solved examples so that you have a better understanding of the concept.
What is the Meaning of Cube Root?
The cube root of a number ‘a’ is a number that is obtained when we multiply three times to get the original number (i.e., the number ‘a’ itself).
Let’s consider the following examples:
If the cube of a number 6, or 63, is equal to 216, then the cube root of 216, or ∛216, is equal to 6.
If the cube of a number 5, or 53, is equal to 125, then the cube root of 125, or ∛125, is equal to 5.
If the cube of a number 4, or 43, is equal to 64, then the cube root of 64, or ∛64, is equal to 4.
In case if we need to find the cube root of any large numbers, then any one of the following methods can be used.
Prime factorization Method
Long Division Method
Using Logarithms
Properties of Cube Root
The cube root of all the odd numbers is an odd number.
For example, ∛125 = 5, ∛27 = 3.
The cube root of all the even natural numbers is even. For example: ∛8 = 2, ∛64 = 4.
The cube root of a negative integer always results in a negative.
What is the Value of Cube Root of 2?
The value of the cube root of 2 is 1.259921 (rounded to 6 decimal places). It is expressed in the form of ∛2. The number 2 is not a perfect cube. Therefore, the general methods cannot be used to find the cube root of 2.
Value of Cube Root of (∛2) = 1.259921
How to Find the Cube Root of 2?
Let’s know how to find the cube root of 2.
If n is a perfect cube for any integer namely m, i.e., n = m3, then m can be known as the cube root of n and it can be denoted by m is equal to ∛n.
Since the number 2 is not a perfect cube, hence we cannot use here the prime factorization method or estimation method to find the cubic root of 2. Therefore, we are going to use the logarithm method to find the cube root of 2..
Using Logarithms
Take log on both the sides,
x = 21/3
So, log(x) = 1/3∗log(2)
log(x) = ⅓∗log(2)
log(x) = ⅓∗0.30102999=0.100343
log(x) = ⅓∗0.30102999=0.100343 (approx)
Therefore,
x = antilog(0.100343)=1.2599
x = antilog(0.100343)=1.2599 (approx)
Value of cube root of 2 =1.2599
Cube Root for Non-perfect Cubes
Finding cube root for non-perfect cubes is not easy. So, we have provided a list of cube roots from 1 to 20 which is composed of only non-perfect cubes.
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 equals to 2 x 7 x 2 x 2 x 7 x 7
Step 2: Form groups of three similar factors, so we get 2 x 2 x 2 x 7 x 7 x 7
Step 3: Take out one factor from each group and multiply.
= 2x 7
= 14
Therefore, \[\sqrt[3]{2744}\] = 14.
Example 2: Find the cube root of 1728 by long division method.
Solution:
Now,
\[\sqrt[3]{1728}=\sqrt[3]{2\times 2\times 2\times 2\times 2\times 2\times 3\times 3\times 3}\]
= 2×2×3
= 12
Example 3: Find the value of ∛2 + ∛(−2).
Solution:
∛−2 = −∛2, which means that the cube root of −2 is equal to the negative of the cube root of 2.
Therefore, ∛2 + ∛(−2) = ∛2 − ∛2 = 0.
FAQs on Understanding the Cube Root of 2 in Radical Form
1. What is the cube root of 2?
The cube root of 2 is the number that, when multiplied by itself three times, equals 2, and its approximate value is ∛2 ≈ 1.2599. In mathematical form:
∛2 = 2^(1/3)
This means:
- 1.2599 × 1.2599 × 1.2599 ≈ 2
- It is a positive real number
- It is not a whole number or a simple fraction
2. Is the cube root of 2 a rational or irrational number?
The cube root of 2 is an irrational number because it cannot be written as a simple fraction of two integers. Its decimal form is non-terminating and non-repeating:
∛2 ≈ 1.259921049…
Since the digits continue infinitely without repeating, it is classified as irrational.
3. How do you calculate the cube root of 2?
The cube root of 2 can be calculated using a calculator, prime factorization (not exact here), or the exponential form 2^(1/3). The simplest method is:
- Enter 2 into a scientific calculator
- Press the cube root (∛) button or raise to the power 1/3
- The result is approximately 1.2599
This value is commonly rounded to four decimal places for practical use.
4. What is the value of cube root of 2 up to 3 decimal places?
The cube root of 2 correct to three decimal places is 1.260. Since:
∛2 ≈ 1.259921…
Rounding to three decimal places gives 1.260 because the fourth decimal digit (9) increases the third digit.
5. What is the cube root of 2 in radical and exponential form?
The cube root of 2 can be written as ∛2 in radical form and 2^(1/3) in exponential form. These two forms are equivalent.
- Radical form: ∛2
- Exponential form: 2^(1/3)
- Decimal approximation: 1.2599
Both representations mean the number that multiplied three times equals 2.
6. Is the cube root of 2 a real number?
Yes, the cube root of 2 is a real number because every positive number has exactly one real cube root. Since 2 is positive, its cube root exists and equals approximately 1.2599.
Unlike square roots of negative numbers, cube roots of real numbers are always real.
7. How do you verify that 1.2599 is the cube root of 2?
You verify it by cubing the number and checking if the result is approximately 2. For example:
- 1.2599 × 1.2599 × 1.2599
- ≈ 2 (rounded)
Since the product is very close to 2, we confirm that ∛2 ≈ 1.2599.
8. What is the difference between √2 and ∛2?
The square root of 2 is √2 ≈ 1.4142, while the cube root of 2 is ∛2 ≈ 1.2599, and they are different values. The difference is:
- √2 × √2 = 2
- ∛2 × ∛2 × ∛2 = 2
Square roots involve power 1/2, while cube roots involve power 1/3.
9. What is the cube root of 2 squared?
The cube root of 2 squared is (∛2)² = 2^(2/3). Using exponent rules:
- ∛2 = 2^(1/3)
- (2^(1/3))² = 2^(2/3)
Its approximate decimal value is ≈ 1.5874.
10. Where is the cube root of 2 used in mathematics?
The cube root of 2 is used in algebra, geometry, and exponential equations where solving cubic equations or working with powers is required. Common applications include:
- Solving equations like x³ = 2
- Simplifying expressions with fractional exponents
- Working in advanced topics like calculus and number theory
It is an important irrational constant in higher-level mathematics.
