Square Root of -192

The square root is the inverse of the square of the number. Since -192 is negative, its square root is not a real number. Instead, it is expressed in terms of the imaginary unit 'i', where i = √-1. Thus, the square root of -192 is expressed as √-192 or 4√12i in simplified form. Since it involves 'i', it is a complex number.

To find the square root of a negative number, we use the concept of imaginary numbers. For -192, we separate it into its positive counterpart and the imaginary unit:

Step 1: Recognize the negative sign in front of 192. This indicates the presence of 'i' in the result, as i = √-1.

Step 2: Find the square root of the positive part, 192, separately, which is √192 = 4√12.

Step 3: Combine this with 'i' to express the full square root: √-192 = 4√12i.

To find the square root of 192, we first perform prime factorization:

Step 1: Finding the prime factors of 192: 192 = 2 x 2 x 2 x 2 x 2 x 3 = 2^5 x 3.

Step 2: Pair the prime factors: (2^5 x 3) can be grouped as 2^2 x 2^2 x 2 x 3.

Step 3: Simplify the pairs: (2^2 x 2^2) becomes 4, and the remaining 2 x 3 remains under the square root.

Thus, √192 = 4√12.

The long division method is not typically used for imaginary numbers, but it can demonstrate finding the square root of the positive part:

Step 1: Group 192 into pairs from right to left: 92 and 1.

Step 2: Find a number whose square is less than or equal to 1. This is 1, with a remainder of 0.

Step 3: Bring down 92, making the new dividend 92. Double the previous quotient (1) to get the new divisor, 2.

Step 4: Find the largest digit 'n' such that 2n x n ≤ 92. In this case, n is 4 because 24 x 4 = 96.

Step 5: Continue the division to find √192.

Step 6: The result is 4√12, which for -192 includes 'i': 4√12i.

The approximation method can be applied to the positive part of -192.

Step 1: Find two perfect squares between which 192 lies. These are 169 (13^2) and 196 (14^2).

Step 2: Since √192 is between √169 and √196, it lies between 13 and 14.

Step 3: Use the approximation formula: (192 - 169) / (196 - 169) = 23/27 ≈ 0.85.

Step 4: Add this to the lower square root: 13 + 0.85 = 13.85.

Step 5: For -192, include 'i': the approximation becomes approximately 13.85i.

• Imaginary unit: The fundamental imaginary unit 'i' is defined as √-1. It is used to handle square roots of negative numbers.

• Complex number: A number that comprises a real and an imaginary part, typically expressed in the form a + bi where 'a' is the real part and 'bi' is the imaginary part.

• Real part: The component of a complex number that does not include the imaginary unit 'i'. For example, in 3 + 4i, the real part is 3.

• Imaginary part: The component of a complex number that includes the imaginary unit 'i'. For example, in 3 + 4i, the imaginary part is 4i.

• Magnitude: The magnitude of a complex number is its absolute value, calculated as the square root of the sum of the squares of its real and imaginary parts.