Final answer:
The smallest whole number to multiply 1458 by to get a perfect square is 6, and the square root of the resulting square number (8748) is 54.
Step-by-step explanation:
The question seeks to find the smallest whole number by which 1458 should be multiplied to obtain a perfect square. To approach this, we need to factorize 1458 and determine its prime factors. We'll then identify which factors need to be squared to create a perfect square and thus find the smallest multiplier.
First, let's factor 1458:
1458 = 2 × 3 × 3 × 3 × 3 × 3 × 3.
We can see that 1458 has a prime factorization with one factor of 2 and six factors of 3. To get a perfect square, each prime factor must appear an even number of times, so we need to multiply 1458 by another 2 to even out the number of 2's and by one more 3 to make the number of 3's even. Therefore, we multiply by 2 × 3 = 6.
The resulting perfect square is 1458 × 6 = 8748.
To find the square root of the new perfect square (8748), we pair the prime factors:
8748 = 2 × 2 × 3 × 3 × 3 × 3 × 3 × 3.
Taking the square root gives us 2 × 3 × 3 × 3 = 54.
Hence, the square root of the square number obtained is 54.