Question

Multiply 6561 by the smallest number so that so that the product is a perfect cube. Also, find the cube root of the product.

Answer 1

Hint: In this question, we are given a number and we have been asked to multiply it with the smallest number so that it results in a perfect cube. In addition to this, we also have to find the number whose perfect cube it is. First, find the prime factors of the number and then, make groups of triplets of prime factors. If the triplets can be made, the number is already a perfect cube, otherwise not. In case it is not a perfect cube, we will see how many numbers it is missing from being a perfect cube and then we will multiply the given number with the missing numbers to get the perfect cube. After that, multiply one number of the triplet with the other number of the other triplet. Do this until all the triplets have been considered and you will get the required cube root.

Complete step-by-step solution:
We will start by factoring the number.
$\Rightarrow 6561=3\times 3\times 3\times 3\times 3\times 3\times 3\times 3$
Now, we will group the factors into triplets.
$\Rightarrow 6561=\left(3\times 3\times 3\right)\times \left(3\times 3\times 3\right)\times 3\times 3$
We could only make 2 triplets and this left behind two 3’s ungrouped. In order to make those two 3’s a triplet, we will multiply $6561$ by a 3. This will result in another triplet and $6561$ will be a perfect cube.
$\Rightarrow 6561\times 3=\left(3\times 3\times 3\right)\times \left(3\times 3\times 3\right)\times \left(3\times 3\times 3\right)$
$\Rightarrow 19683=\left(3\times 3\times 3\right)\times \left(3\times 3\times 3\right)\times \left(3\times 3\times 3\right)$
We have our perfect cube. Now, we will find the cube root.
$\Rightarrow \sqrt[3]{19683}=?$
We have three triplets here. Hence, we will multiply 3 - three times to find the cube root.
$\Rightarrow \sqrt[3]{19683}=3\times 3\times 3=27$

Hence, the cube root of 19683 is 27.

Note: There are several other ways to make a triplet of 6561 –
We could have also divided 6561 by 9. This would have given us a cube of 9.
We could have also multiplied 6561 by 81. This would have given us a cube of 81.
But we didn’t choose any of the above methods because it was not demanded in the question.