Final answer:
The simplified form of the square root of negative 54 is 3i√6, as it involves the imaginary unit i and factorization of 54 into a perfect square (9) and 6.
Step-by-step explanation:
The student asked what is the simplified form of the square root of negative 54. To find this, we first need to address that the square root of a negative number involves the imaginary unit i, where i = √-1. Numbers of the form a+bi are part of the complex number system, and here we are dealing with a purely imaginary number because our real part is 0.
First, we will factor out -1 from the square root to deal with the negative sign: √-54 = √(-1*54)
√(-1*54) = √-1 * √54 = i√54
Then we look for perfect squares in 54: 54 = 9*6, and 9 is a perfect square.
Break down 54 into its prime factors: √54 = √(9*6) = √9 * √6
The square root of 9 is 3, thus we have: 3√6
Combining this with i, we get: 3i√6
Therefore, the simplified form of the square root of negative 54 is 3i√6, which corresponds to option A.