0.5*(632-189) = 1890 diagonals
5607 + 18 = 5625, a perfect square. The perfect square of a square root is the number you started with.
300*3 = 900 which is a perfect square because 30 squared equals 900
62
To find the smallest positive integer ( n ) such that ( 2n ) is a perfect square, ( 3n ) is a perfect cube, and ( 4n ) is a perfect fourth, we analyze the conditions for each case using prime factorization. Let ( n = 2^a \cdot 3^b \cdot k ), where ( k ) is coprime to 2 and 3. For ( 2n ) to be a perfect square, ( a+1 ) must be even and ( b ) must be even. For ( 3n ) to be a perfect cube, ( a ) must be divisible by 3 and ( b+1 ) must be divisible by 3. For ( 4n ) to be a perfect fourth, ( a+2 ) must be divisible by 4 and ( b ) must be divisible by 4. By solving these conditions simultaneously, the smallest ( n ) that meets all conditions is ( n = 108 ).
To find the least number that should be added to 81180 to make it a perfect square, first, determine the square root of 81180, which is approximately 284.3. The next whole number is 285, and squaring it gives 81225. Therefore, the least number to be added is 81225 - 81180 = 45. Thus, 45 should be added to 81180 to make it a perfect square.
If its square root can be expressed as a rational number then it is a perfect square. 9075 is not a perfect square. However, 9025 is.
13
5
-- Find the square root of 4,321.-- It begins with 65.7...-- So the smallest perfect square greater than 4,321 must be (66)2.-- (66)2 = 4,356 .-- 4,356 - 4,321 = 35 .
To determine the least number that must be multiplied to 21168 to make it a perfect square, we first find its prime factorization: (21168 = 2^4 \times 3^1 \times 11^1). For a number to be a perfect square, all the exponents in its prime factorization must be even. Here, the exponent of 3 and 11 are odd, so we need at least one more factor of each. Thus, we multiply by (3^1 \times 11^1 = 33). Therefore, the least number that must be multiplied to 21168 to make it a perfect square is 33.
100015
To find the least whole number by which 2100 can be divided to make the result a perfect square, we need to factorize 2100. The prime factorization of 2100 is 2^2 * 3 * 5^2 * 7. To make it a perfect square, we need to pair up the prime factors. We see that if we pair up 2 * 5 * 7 = 70, the result is a perfect square. Therefore, the least whole number by which 2100 can be divided to make the result a perfect square is 70.