Question

Find the nearest integer to the cube root of 12345.

Answer 1

Hint: In this question, we need to use the trial and error method of guessing the numbers and their cube, to find the number closer to the cube root of 12345. Use $ {{a}^{3}}=a\times a\times a $ to find the cube of any particular number.

Complete step-by-step answer:
Here we have to find the nearest integer to the cube root of 12345.
A cube root of a number is a value which has been multiplied three times gives that number.
We do not know if 12345 is a perfect cube so we will guess the cube of the numbers closer to the number 12345.
We know,
 $ {{a}^{3}}=a\times a\times a $
Now let us take the cube of number 20, we get
 $ {{20}^{3}}=20\times 20\times 20 $
Here, we need to multiply all the three 20s’ to get the value of the cube of 20.
 $ {{20}^{3}}=8000 $
8000 is way smaller than 12345. Therefore, let us take the cube of a number 25.
 $ \begin{align}
 & {{25}^{3}}=25\times 25\times 25 \\
 & =625\times 25 \\
 & =15625
\end{align} $
Here, the number 15625 is bigger than 12345.
So, now we can say that the number is between the numbers 20 and 25.
Let us take the numbers 24 and 23 and find their cubes, in order to see which number is closer to the number 12345.
For 24,
 $ \begin{align}
 & {{24}^{3}}=24\times 24\times 24 \\
 & =13824
\end{align} $
For 23,
$ \begin{align}
& {{23}^{3}}=23\times 23\times 23 \\
& =12167
\end{align} $
12167 < 12345 < 13824
This means that the number 12345 lies between the cube of 23 and the cube of 24.
Therefore, $ {{\left( 23 \right)}^{3}}<12345<{{\left( 24 \right)}^{3}} $ .
But if we observe closely, we can see that 12167 is closer to 12345 than 13824.
Hence, the nearest integer to the cube root of 12345 is number 23.

Note: Here, instead of a trial and error method, you can calculate the cube root by using the division method for cube roots and then write the number which is closer to the cube root of the number. For example, the cube root of 12345 = 23.11, here the decimal is closer to 23 than 24, hence the nearest integer to the cube root is 23.