How to Find the Cube Root of 4 Step by Step with Formula and Examples
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 Cube Root of 4 Explained with Meaning and Value
1. What is the cube root of 4?
The cube root of 4 is the number that, when multiplied by itself three times, equals 4, and its approximate value is ∛4 ≈ 1.5874. In mathematical form, ∛4 = 4^(1/3). Since 4 is not a perfect cube, its cube root is an irrational number with non-terminating, non-repeating decimals.
2. How do you calculate the cube root of 4?
The cube root of 4 can be calculated using a calculator, prime factorization, or exponential form as 4^(1/3).
- Step 1: Write 4 as 4^(1/3).
- Step 2: Use a scientific calculator and enter 4 followed by the cube root function.
- Step 3: The result is approximately 1.5874.
3. Is the cube root of 4 a rational or irrational number?
The cube root of 4 is an irrational number because it cannot be expressed as a simple fraction. Its decimal form, approximately 1.587401..., is non-terminating and non-repeating. Since 4 is not a perfect cube, its cube root is not a whole number or rational fraction.
4. What is the value of cube root of 4 in radical form?
The value of the cube root of 4 in radical form is written as ∛4. This is already the simplest radical form because 4 has no perfect cube factors other than 1. It can also be expressed in exponential form as 4^(1/3).
5. What is 4 raised to the power 1/3?
The expression 4^(1/3) is equal to the cube root of 4, which is approximately 1.5874. Fractional exponents follow the rule a^(1/n) = ∛a (for n = 3), so 4^(1/3) and ∛4 represent the same value.
6. Is the cube root of 4 a perfect cube?
No, 4 is not a perfect cube because there is no integer whose cube equals 4. Perfect cubes include numbers like 1 (1³), 8 (2³), and 27 (3³). Since 4 lies between 1 and 8, its cube root lies between 1 and 2.
7. How can you estimate the cube root of 4 without a calculator?
You can estimate the cube root of 4 by comparing nearby perfect cubes.
- 1³ = 1
- 2³ = 8
- 1.5³ = 3.375
- 1.6³ = 4.096
8. What is the cube root of 4 rounded to three decimal places?
The cube root of 4 rounded to three decimal places is 1.587. The exact decimal starts as 1.587401..., and rounding to three decimal places gives 1.587.
9. What is the difference between the square root and cube root of 4?
The square root of 4 is 2, while the cube root of 4 is approximately 1.5874.
- Square root means a number multiplied by itself twice (2 × 2 = 4).
- Cube root means a number multiplied by itself three times (1.5874 × 1.5874 × 1.5874 ≈ 4).
10. What is the cube root of 4 in simplest exponential form?
The cube root of 4 in simplest exponential form is 4^(1/3). Using the law of fractional exponents, a^(1/n) represents the nth root of a. Therefore, 4^(1/3) is exactly equal to ∛4 and approximately equal to 1.5874.
