Question

The least number that should be multiplied with 8 power 1729 into 6 power 1947 to make it a perfect square

Answer 1

Answer:

Step-by-step explanation:8 power 1729×6 power 1947  

81729×21947×31947

23(1729)×21947×31947

25187×21947×31947

25187+1947×31947

27134×31947

To make it perfect square multiply with “3”

Answer 2

Find the smallest number when multiplied by $8^{1729} 6^{1947}$ makes it a perfect square.

Explanation:

• For a composite number to be a perfect square the powers of each of its prime factors should be even i.e, multiple of two.
• Now we have the higher powers of numbers eight and six.
• Hence, the prime factorization of eight and six is,
•                   $->8=2(2)(2)=2^3 \\->6=2(3)$    
• Now, the required higher power of them gives us,
•                   $->8^{1729}=2^{3(1729)}=2^{5187} \\->6^{1947}=2^{1947}3^{1947} \\->8^{1729} 6^{1947}= 2^{5187+1947}3^{1947}\\->8^{1729} 6^{1947}= 2^{7134}3^{1947}$  
• Hence the power of two is already even but the power of three is odd.
• Therefore, in order to make the power of three even multiply it by three,  $->3(8^{1729} 6^{1947})= 2^{7134}3^{1948}= (2^{3567}3^{974})^2$  
• The least number to make the given number a perfect square is $3$.