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What do we Perceive by the Cube Root of 4?

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Last updated date: 17th Apr 2024
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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 Difference Between the Square Root and Cube Root?

  • The square root of a number x is that number which when multiplied by itself gives the number x itself. For example, the square root of 4 is 2.

  • The cube root of a number is a number that when multiplied three times gives you the number. For example, the cube root of 216 is 6.

  • The cube root of a number is the length of a cube whose volume is \[a^{3}\] units. Whereas the square root of a number is the length of a square whose area is \[a^{2}\] units.

  • The square root is an inverse operation of squaring a number, while the cube root is an inverse operation of cubing a number.

  • The square root is dented by  \[\sqrt{a}\], while the cube root is denoted by  \[3 \sqrt{a}\].

2. What is a Square Root?

The square root of a number x is that number which when multiplied by itself gives the number x itself. The number x is a perfect square.

For Example, \[ 2^{2} = 4\], or the square root of 4 is 2

\[ 3^{9} = 4\], or the square root of 9 is 3

\[ 4^{2} = 16\], or the square root of 16 is 4

The symbol of the square root is \[\sqrt{}\] 

Therefore, the square root of 4 is represented as \[\sqrt{4} = 2\].

And the square root of 9 is represented as \[\sqrt{9} = 3\] and so on.