Question

check if 5832 is a perfect cube by prime factorisation method​

Answer 1

Answer:

Complete step-by-step solution -

Here, we have to find the cube root of 5832.

We know that a cube root of a number is a special value that, when used in a multiplication three times, gives that number. That is, if ‘a’ is the cube root of x then we can write x=a3=a×a×a.

For example, 23=2×2×2=8, so the cube root of 8 is 3.

Here, to find the cube root we can use the method of prime factorisation.

In prime factorisation we factorise the numbers into prime numbers, called as prime factors. We have to take the following steps for prime factorisation.

1. First we divide the number by the smallest prime number which divides the number exactly.

2. We divide the quotient again by the smallest prime number, or the next smallest prime number, if it is not exactly divisible by the smallest prime number. Repeat the process again and again till the quotient becomes 1. Here we are using only prime numbers to divide.

3. We multiply all the prime factors, where the product is the number itself.

Therefore, for finding cube root by prime factorisation we have to group the factors in three such that each number of the group is the same. Take one factor from each group and then multiply to obtain the cube root. Now, let us do the prime factorisation of 5832.

Now, by prime factorisation 5832 can be written as:

5832=2×2×2×3×3×3×3×3×3

Next, we have to group the numbers into three with each group having the same members. Hence we will get:

5832=(2×2×2)×(3×3×3)×(3×3×3)

Now, take one factor from each group, and take cube root of 5832, we will obtain:

5832−−−−√3=2×3×3

In the next step we have to multiply these factors. The number so obtained will be the required cube root. Therefore, we can write:

5832−−−−√3=18

So, we got 5832 as the perfect cube of 18.

Hence, we can say that the cube root of 5832 is 18.

Note: Here, to find the prime factors using repetitive division it is always better to start the division with a smaller prime factor and continue the process with bigger prime factors. If you start with a bigger number there is a chance of division going wrong.

Answer 2

18

We have to take the following steps for prime factorisation.

1. First we divide the number by the smallest prime number which divides the number exactly.

2. We divide the quotient again by the smallest prime number, or the next smallest prime number, if it is not exactly divisible by the smallest prime number. Repeat the process again and again till the quotient becomes 1. Here we are using only prime numbers to divide.

3. We multiply all the prime factors, where the product is the number itself.

Therefore, for finding cube root by prime factorisation we have to group the factors in three such that each number of the group is the same. Take one factor from each group and then multiply to obtain the cube root. Now, let us do the prime factorisation of 5832.

Now, by prime factorisation 5832 can be written as:

5832 = 2×2×2×3×3×3×3×3×3

Next, we have to group the numbers into three with each group having the same members. Hence we will get:

5832 = (2×2×2)×(3×3×3)×(3×3×3)

Now, take one factor from each group, and take cube root of 5832, we will obtain:

$\sqrt[3]{5832} = 2 \times 3 \times 3$

In the next step we have to multiply these factors. The number so obtained will be the required cube root. Therefore, we can write

$\sqrt[3]{5832} = 18$

So, we got 5832 as the perfect cube of 18.

Hence, we can say that the cube root of 5832 is 18.