Question

Examine if $398$ is a perfect cube. If not, then find the smallest number that must be subtracted to obtain a perfect cube.

Answer 1

Hint: In order to solve this question, we need to find the perfect cube which is nearest to the number $398$. This can be done by finding the cubes of every natural number starting from $1$.
Once you obtain the nearest perfect cube subtract it from$398$ to get your answer.

Complete step-by-step answer:
First, we need to check whether $398$ is a perfect cube or not.
For that, we need to find the cubes of the natural number starting from $1$
Cube of $1 = 1 \times 1 \times 1 = 1$
Cube of $2 = 2 \times 2 \times 2 = 8$
Cube of $3 = 3 \times 3 \times 3 = 27$
Cube of $4 = 4 \times 4 \times 4 = 64$
Cube of $5 = 5 \times 5 \times 5 = 125$
Cube of $6 = 6 \times 6 \times 6 = 216$
Cube of $7 = 7 \times 7 \times 7 = 343$
Cube of $8 = 8 \times 8 \times 8 = 512$
As we can see that none of the cubes until $8$ has the value $398$ and from here the value of cubes will increase.
Therefore $398$ is not the perfect cube.
Now if you read the question carefully, it is written that we must subtract the number from $398$ in order to get the nearest cube which means that we need to get the cube smaller than $398$
As we can see that ${8^3} = 512$ which is greater than $398$ and ${7^3} = 343$ which is less than $398$ are the two perfect cubes from $398$
As we need the smaller perfect cube, hence the perfect cube will be taken as ${7^3} = 343$
Now we need to find the number that must be subtracted from $398$ to get $343$.
Let that number be $x$
$\begin{gathered}
  398 - x = 343 \\
  x = 55 \\
\end{gathered} $
Hence $55$ must be subtracted from $398$ to get the nearest perfect cube.

Note: This question can also be solved by taking the cube root of $398$ and the cube root of it is
$7.3557623684$ and as it is in decimals then we can say that it is not a perfect cube. Since the nearest perfect cube is to be smaller, therefore it must be the cube of $7$