Q. 75.0( 4 Votes )

Let us write the

Answer :

Let us recall the definition of prime factors and cube roots.

Prime factors are any of the prime numbers that can be multiplied to give the original number.


The cube root of a number is a special value that, when used in a multiplication three times, gives that number.


(i). For 512:


512 can be divided by prime number 2, quotient is 256.


256 can be divided by prime number 2, quotient is 128.


128 can be divided by prime number 2, quotient is 64.


64 can be divided by prime number 2, quotient is 32.


32 can be divided by prime number 2, quotient is 16.


16 can be divided by prime number 2, quotient is 8.


8 can be divided by prime number 2, quotient is 4.


4 can be divided by prime number 2, quotient is 2.


2 can be divided by prime number 2, quotient is 1.



So, 512 can be expressed in product of prime factors as:


512 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2


Group these factors into three similar integers.


512 = (2 × 2 × 2) × (2 × 2 × 2) × (2 × 2 × 2)


Taking cube root on both sides, we get





Thus, cube root of 512 is 8.


(ii). For 1728.


1728 can be divided by prime number 2, quotient is 864.


864 can be divided by prime number 2, quotient is 432.


432 can be divided by prime number 2, quotient is 216.


216 can be divided by prime number 2, quotient is 108.


108 can be divided by prime number 2, quotient is 54.


54 can be divided by prime number 2, quotient is 27.


27 can be divided by prime number 3, quotient is 9.


9 can be divided by prime number 3, quotient is 3.


3 can be divided by prime number 3, quotient is 1.



So, 1728 can be expressed in product of prime factors as:


1728 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3


Group these factors into three similar integers.


1728 = (2 × 2 × 2) × (2 × 2 × 2) × (3 × 3 × 3)


Taking cube root on both sides, we get





Thus, cube root of 1728 is 12.


(iii). For 5832.


5832 can be divided by prime number 2, quotient is 2916.


2916 can be divided by prime number 2, quotient is 1458.


1458 can be divided by prime number 2, quotient is 729.


729 can be divided by prime number 3, quotient is 243.


243 can be divided by prime number 3, quotient is 81.


81 can be divided by prime number 3, quotient is 27.


27 can be divided by prime number 3, quotient is 9.


9 can be divided by prime number 3, quotient is 3.


3 can be divided by prime number 3, quotient is 1.



So, 5832 can be expressed in product of prime factors as:


5832 = 2 × 2 × 2 × 3 × 3 × 3 × 3 × 3 × 3


Group these factors into three similar integers.


5832 = (2 × 2 × 2) × (3 × 3 × 3) × (3 × 3 × 3)


Taking cube root on both sides, we get





Thus, cube root of 5832 is 18.


(iv). For 15625.


15625 can be divided by prime number 5, quotient is 3125.


3125 can be divided by prime number 5, quotient is 625.


625 can be divided by prime number 5, quotient is 125.


125 can be divided by prime number 5, quotient is 25.


25 can be divided by prime number 5, quotient is 5.


5 can be divided by prime number 5, quotient is 1.



So, 15625 can be expressed in product of prime factors as:


15625 = 5 × 5 × 5 × 5 × 5 × 5


Group these factors into three similar integers.


15625 = (5 × 5 × 5) × (5 × 5 × 5)


Taking cube root on both sides, we get





Thus, cube root of 15625 is 25.


(v). For 10648.


10648 can be divided by prime number 2, quotient is 5324.


5324 can be divided by prime number 2, quotient is 2662.


2662 can be divided by prime number 2, quotient is 1331.


1331 can be divided by prime number 11, quotient is 121.


121 can be divided by prime number 11, quotient is 11.


11 can be divided by prime number 11, quotient is 1.



So, 10648 can be expressed in product of prime factors as:


10648 = 2 × 2 × 2 × 11 × 11 × 11


Group these factors into three similar integers.


10648 = (2 × 2 × 2) × (11 × 11 × 11)


Taking cube root on both sides, we get





Thus, cube root of 10648 is 22.


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