# LUCAS WILLEMS

A 23 year-old student passionate about maths and programming

A 23 year-old student passionate about maths and programming

I was watching maths videos on Youtube when I came accross this video of Preety Uzlain explaining how to find cube root of cube of 2-digits numbers.

So, if we want to find the value of \(b = \sqrt[3]{a}\) where \(a\) is the cube of a 2-digits number, we have to use the following technique :

- Find what is the number \(u \in [[0, 9]]\) that the cube has the same units digit as \(a\)
- Find what is the number \(d \in [[0, 9]]\) that the cube is less than \(\left \lfloor \frac{a}{1000} \right \rfloor\) and the nearest of this value

When \(u\) and \(d\) are found, we conclude that \(b = 10d + u\).

The purpose of this article is to show you that Preety Uzlain's technique to calculate cubic root of cube of 2-digits numbers works.

As we want to find \(\sqrt[3]{a}\) where \(a\) is the cube of 2-digits number, we can write :

$$(d, u) \in [[0, 9]]^2 \qquad a = (10d + u)^3$$

Let's develop the previous expression :

$$\begin{align}a &= (10d)^3 + 3(10d)^2 u + 3(10d)u^2 + u^3 = 1000d^3 + 300d^2 u + 30du^2 + u^3 \\ &= 10(100d^3 + 30d^2 u + 3du^2) + u^3\end{align}$$

It is now easy to see that the units digit of \(10(100d^3 + 30d^2 u + 3du^2)\) is \(0\), which means that \(a\) and \(u^3\) have the same units digit.

So, as the units digit of \(0^3\), \(1^3\), \(2^3\), ..., \(9^3\) is different, only 1 \(u^3\) with the same digits unit as \(a\) exists. It is this reason that proves the 1st point of the technique.

As \(0 \leqslant u < 10\), we get the following inequality :

$$10d \leqslant 10d + u < 10d + 10 = 10(d+1)$$And, if we divide by 10 then cube each member of the inequality, we get :

$$d^3 \leqslant \frac{(10d + u)^3}{1000} < (d+1)^3$$

As \(a = (10d + u)^3\), the final inequality is the following :

$$d^3 \leqslant \frac{a}{1000} < (d+1)^3$$It is now very easy to see that the 2nd point of the technique is right.

Therefore, the technique used by Preety Uzlain works !

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- Proof calculation of some cubic roots
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