Cube root of 4 – Different methods

The cube root of 4 is 1.5874 (up to 4 decimal places). In this article, we are going to find out its value using two different methods.

What do you mean by Cube root of 4?

When a number is multiplied by itself three times to give a product as 4, then that number is the cube root of 4.

It can be also written as \sqrt[3]{4} or 4^{1 / 3}.

After the prime factorization of 4, we get

4 = 2 × 2

Since factor 2 is not in the group of three, 4 is not a perfect cube.

\sqrt[3]{4} is an irrational number as it cannot be expressed in the form of  \frac{p}{q},  where p and q are integers.

Finding the value of \sqrt[3]{4}

Since 4 is not a perfect cube, we cannot use the prime factorization or estimation method to find its cube root.

Therefore, we have to follow the trial and error method to find the value of \sqrt[3]{4}.

There are some other methods such as Halley’s methods and Newton Raphson method. But we will study about these in higher classes.

Trial 1: 

4 lies between the perfect cubes 1 and 8.

Hence, the cube root of 4 lies between cube root of 1 and cube root of 8.

1<\sqrt[3]{4}<2

Trial 2: 

We can write 4 as  \frac{4000}{1000}

So, \sqrt[3]{4}=\frac{\sqrt[3]{4000}}{\sqrt[3]{1000}}=\frac{\sqrt[3]{4000}}{10}.

4000 is also not a perfect cube and it lies between perfect cubes 3375 and 4096.

Hence the cube root of 4000 lies between cube root of 3375 and cube root of 4096.

15<\sqrt[3]{4000}<16

\Rightarrow \quad \frac{15}{10}<\frac{\sqrt[3]{4000}}{10}<\frac{16}{10}

\Rightarrow 1.5<\sqrt[3]{4}<1.6

Trial 3: 

We can write 4 as \frac{4000000}{1000000}.

So, \sqrt[3]{4}=\frac{\sqrt[3]{4000000}}{\sqrt[3]{1000000}}=\frac{\sqrt[3]{4000000}}{100}

4000000 is not a perfect cube and it lies between the perfect cubes 3944312 and 4019679.

Hence the cube root of 4000000 lies between the cube root of 3944312 and the cube root of 4019679.

158<\sqrt[3]{4000000}<159

\Rightarrow \quad \frac{158}{100}<\frac{\sqrt[3]{4000000}}{100}<\frac{159}{100}

\Rightarrow 1.58<\sqrt[3]{4}<1.59

Since 4000000 is much closer to 4019679 we can assume that the approximate value of \sqrt[3]{4}rounded up to 2 decimal places is 1.59.

\Rightarrow \quad \sqrt[3]{4} \approx 1.59

This process is time-consuming but gives values much nearer to the actual value.

Finding the value of \sqrt[3]{4} when we know the value of \sqrt[3]{2}

We can write 4 as \frac{8}{2}.

Here the numerator 8 is a perfect cube.

\sqrt[3]{4}=\frac{\sqrt[3]{8}}{\sqrt[3]{2}}=\frac{2}{\sqrt[3]{2}}

If we know that the approximate value of \sqrt[3]{2} is 1.26

Therefore, \sqrt[3]{4}=\frac{2}{\sqrt[3]{2}} \approx \frac{2}{1.26} \approx 1.5873

This value of 1.5873 is also much closer to the actual value of 1.5874. But for this method, we need to remember the value of \sqrt[3]{2}.

Solved Examples:

1. Find the approximate value of cube root of 108?

Solution

We know, 108 = 4 × 27

Taking Cube Root on both sides

\sqrt[3]{108}=\sqrt[3]{(4 \times 27)}

\Rightarrow \sqrt[3]{108}=\sqrt[3]{(4)} \times \sqrt[3]{(27)}

We already know that 3 is the cube root of 27 and 1.587 is the approximate value of cube root of 4.

Therefore,\sqrt[3]{108}=1.587 \times 3

\Rightarrow \sqrt[3]{108}=4.761