Let’s get a closer look to understand!
The value when multiplied by itself three consecutive times gives the result in the original number. It is denoted as \[\sqrt[3]{4}\]. The cube root is defined as the root of a number that can be generated by taking the cube of another number. Thus, if the value of \[\sqrt[3]{4} = x\], then \[x^{3} = 4\] and we need to determine the value of x.
Here is a detailed explanation of how to find the cube root of 4.
Cube root of 4, \[\sqrt[3]{4} = 1.5874\]
Let’s Imbibe The Manner To determine the value of cube root of 4
Since the number '4' is not a perfect cube, the prime factorization method or estimation method cannot be used to determine the cubic root of 4. In this case, we can use another method to determine the cube root of '4' that is Halley's method. The following are the steps to determine the cube root of '4' using Halley’s Method.
The formula for Halley’s method is,
\[\sqrt[3]{a} \times (\frac{(x^{3} + 2a)}{(2x^{3} + a)})\]
Where,
'a' is the number to which the cube root is to be calculated.
'x' is the integer guess of the cube root.
Here, to determine the cube root of 4; where a = 4.
Let us take a = 1 since 1 is the closest perfect cube that is less than 4.
(i.e) \[1^{3} = 1\].
Therefore, x = 1.
If we substitute the values using the Halley’s formula, we get
\[\sqrt[3]{4} = 1 \left ( \frac{13 + 2 \times 4}{2 \times 13 + 4} \right )\]
\[\sqrt[3]{4} = 1.5 \]
Hence, the cube root of 4 is more or less equal to 1.5.
Examples Related to Cube Root of 4
Example 1: Simplify \[4 + \sqrt[3]{4}\].
Solution:
Given: \[4 + \sqrt[3]{4}\].
The value of cube root of 4 or \[\sqrt[3]{4}\] is approximately equal to 1.5874.
By substituting the cube root value of 4 in the given expression, we get,
\[4 + \sqrt[3]{4}\] = 4+1.5874.
\[4 + \sqrt[3]{4}\] = 5.5874.
Hence, \[4 + \sqrt[3]{4}\] = 5.5874.
Example 2: Determine the value of x for the equation: \[x + 2\sqrt[3]{4}\] = 10.
Solution:
Given expression: \[x + 2\sqrt[3]{4}\] = 10
As we know that, \[\sqrt[3]{4}\] = 1.5874.
By substituting the cube root value of 4 in the given equation, we get
\[x + 2\sqrt[3]{4}\] = 10
x+2(1.5874) = 10
x+ 3.1748 = 10
x = 10-3.1748
x = 6.8252.
Hence, the value of x is 6.8252.
Conclusion:
Cube Roots are important to solve many mathematical equations. The conceptual understanding is of utmost importance here. If you decipher the tricks around cube roots, getting to the solution becomes a less tedious exercise.
FAQs on What do we Perceive by the Cube Root of 4?
1. What is the value of the cube root of 4?
The cube root of 4, denoted as ∛4 or 4¹/³, is an irrational number. Its value is approximately 1.5874. This is the number that, when multiplied by itself three times (1.5874 × 1.5874 × 1.5874), results in a value very close to 4.
2. What is the symbol used to represent the cube root of 4?
The symbol for the cube root is the radical sign (√) with a small '3' (called the index) placed in the crook of the symbol. Therefore, the cube root of 4 is written as ∛4. It can also be expressed in exponential form as 4¹/³.
3. How can we calculate or estimate the cube root of 4?
Since 4 is not a perfect cube, its cube root cannot be found by simple factorisation. We can estimate its value by a process of approximation:
- We know that 1³ = 1 and 2³ = 8.
- Since 4 lies between 1 and 8, its cube root must be a value between 1 and 2.
- For a precise value, methods like the long division method for cube roots or using a scientific calculator are necessary. This gives an approximate value of 1.5874.
4. How is the 'cube of 4' different from the 'cube root of 4'?
These are inverse mathematical operations that yield very different results.
- The cube of 4 involves multiplying 4 by itself three times: 4 × 4 × 4, which equals 64.
- The cube root of 4 (∛4) is the number that, when cubed, equals 4. Its value is approximately 1.5874.
5. What is the difference between the square root and the cube root of a number?
The primary difference lies in the root index, which dictates how many times a number must be multiplied by itself.
- Square Root (√): Represents a number that, when multiplied by itself once (e.g., y × y), equals the original number. For example, the square root of 4 is 2 because 2 × 2 = 4.
- Cube Root (∛): Represents a number that, when multiplied by itself twice (e.g., y × y × y), equals the original number. For example, the cube root of 8 is 2 because 2 × 2 × 2 = 8.
6. Can the cube root of 4 be expressed as a simple fraction?
No, the cube root of 4 cannot be written as a simple fraction (in the form a/b where a and b are integers). This is because ∛4 is an irrational number. Its decimal form is non-terminating and non-repeating (1.58740105...), meaning any fractional representation would only be an approximation, not an exact value.
7. How can the cube root of 4 be simplified?
The cube root of 4, written as ∛4, is already in its simplest radical form. To simplify a cube root, one must find a perfect cube factor within the number under the radical. The prime factorisation of 4 is 2 × 2. Since there is no group of three identical factors, no integer can be moved outside the radical sign. Therefore, ∛4 cannot be simplified further.
8. Why is the cube root of 4 considered an irrational number?
The cube root of an integer is rational only if that integer is a perfect cube. A perfect cube is a number obtained by multiplying an integer by itself three times (e.g., 1³=1, 2³=8, 3³=27). Since 4 is not a perfect cube, its cube root cannot be a whole number or a terminating/repeating decimal. This property classifies ∛4 as an irrational number.
