what is the index of three with the square root of negative 432
To find the index of three with the square root of negative 432, we first need to determine the square root of negative 432. However, it's important to note that the square root of a negative number results in an imaginary number.
Since we're dealing with an imaginary number, we can write it in the form of "a + bi", where "a" represents the real part and "b" represents the imaginary part. For example, the imaginary square root of -1 is written as "i".
Let's calculate the square root of negative 432:
√(-432) = √(432) * √(-1) = √(432)i
Now, to find the index of three with the square root of negative 432, we need to raise it to the power of three.
(√(432)i)^3 = (√(432))^3 * i^3 = 6^3 * i^3 = 216 * i^3
To simplify further, we need to determine the value of i^3.
The powers of i cycle in sets of four: i, -1, -i, 1. To find i^3, we can calculate i^3 modulo 4, which refers to the remainder when i^3 is divided by 4.
Since i^3 is equivalent to i * i * i, we can see that i^3 divides evenly by 4 with no remainder. Therefore, i^3 equals 1.
Hence, (√(432)i)^3 simplifies to 216 * 1, which is equal to 216.
Therefore, the index of three with the square root of negative 432 is 216.