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 ).
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
The sqrt. will be the length of one side.
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.