Is 9720 a perfect cube if not find the smallest number by which should be divided to get a perfect cube?

Given: 9720

Factors of 9720 = 2 × 2 × 2 × 3 × 3 × 3 × 3 × 3 × 5

As we see, 3 and 5 do not have a pair to make a cube.

Hence 9720 should be divided with 3 × 3 × 5 to get perfect cube.

⇒ 9720 should divide with 45.

Now, after division we get 216 = 63

Now, 216 is a perfect cube.

First we have to find out the factors by using prime factorisation method.

So, prime factors of 9720 = 2 × 2 × 2 × 3 × 3 × 3 × 3 × 3 × 5

Now, grouping the prime factors = (2 × 2 × 2) × (3 × 3 × 3) × 3 × 3 × 5

= 23 × 33 × 3 × 3 × 5

Factors 3 and 4 has no pair.

∴9720 is not a perfect cube.

The smallest number it should be divided to get a perfect cube is 3 × 3 × 5 = 45.

Then,

= 9720 ÷ 45

= 216

Factors of 216 = (2 × 2 × 2) × (3 × 3 × 3)

3√216 = 3√((2 × 2 × 2) × (3 × 3 × 3))

= 3√(23 × 32)

= 6

Solution:

Given, the number is 9720.

We have to determine if 9720 is a perfect cube or not.

Prime factorization is a way of expressing a number as a product of its prime factors.

A prime number is a number that has exactly two factors, 1 and the number itself.

Using prime factorisation,

So, 9720 = 2 × 2 × 2 × 3 × 3 × 3 × 3 × 3 × 5

We observe that 3 and 5 occur without pairs.

Therefore, 9720 is not a perfect cube.

We have to determine the smallest number by which 9720 should be divided to get a perfect cube.

9720 should be divided by 3 × 3 × 5

3 × 3 × 5 = 45

So, 9720/45 = 216

Factors of 216 = 2 × 2 × 2 × 3 × 3 × 3

Therefore, the smallest number to be divided is 45.

✦ Try This: Is 930 a perfect cube? If not, find the smallest number by which it should be divided to get a perfect cube.

☛ Also Check: NCERT Solutions for Class 8 Maths

NCERT Exemplar Class 8 Maths Chapter 3 Problem 99

Is 9720 a perfect cube? If not, find the smallest number by which it should be divided to get a perfect cube

Summary:

9720 is not a perfect cube. The smallest number by which 9720 should be divided to get a perfect cube is 45.

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• Write two Pythagorean triplets each having one of the numbers as 5
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Is 9720 a perfect cube? If not, find the smallest number by which it should be divided to get a perfect cube.

Answer

Verified

Hint: To check whether given number 9720 is a perfect cube or not, we will find its factor by finding its least common factor. We check for the triplets formed by the factors of the number. If all the factors form a triplet, then the number can be categorised as a perfect cube.
If not, then we will check for the number which are not forming a triplet. To get a number which is a perfect cube, we will divide the given number by the numbers which are not forming triplets in the factors.

Complete step by step solution
To check whether the given number 9720 is a perfect cube or not, we find will find its factor by finding its least common factor which can be expressed as:
$\begin{array}{l}
9720 = 2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 3 \times 3 \times 5\\
9720 = \underline {2 \times 2 \times 2} \times \underline {3 \times 3 \times 3} \times 3 \times 3 \times 5
\end{array}$

Since, we can see that there is no triplet for 3 and 5 in the above expression, hence the number 9720 is not a perfect cube.

 We can see that $3 \times 3 \times 5$ or 45 is the number because 9720 is not a perfect cube.

Hence, we will divide our number with $45\left( {3 \times 3 \times 5} \right)$ we can get the number which is a perfect cube. Since we will be left with two triplets. This can be expressed as:
$\begin{array}{l}
\dfrac{{9720}}{{45}} = \dfrac{{2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 3 \times 3 \times 5}}{{3 \times 3 \times 5}}\\
216 = \underline {2 \times 2 \times 2} \times \underline {3 \times 3 \times 3} \\
216 = {\left( {2 \times 3} \right)^3}\\
216 = {6^3}
\end{array}$

As 216 is the perfect cube, therefore, 45 is the smallest number by which the number 9720 should be divided to get a perfect cube.

Note: To check whether a number is a perfect square or not, we have two methods, one is through a long division method and other is by finding its factors. But to check whether a number is a perfect cube or not, we will only check for triplets in the factors.

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