Square Root of -104

The square root is the inverse of the square of a number. Since -104 is a negative number, its square root is a complex number. The square root of -104 can be expressed in terms of the imaginary unit i, where i² = -1. Therefore, the square root of -104 is expressed as √-104 = √(104) × i = 10.198 × i, which is a complex number.

To find the square root of a negative number, the concept of complex numbers is used. The steps involve finding the square root of the absolute value and multiplying by the imaginary unit i.

Step 1: Find the square root of the absolute value of -104, which is 104.

Step 2: √104 = 10.198.

Step 3: Multiply the result by i to get the square root of -104: √-104 = 10.198 × i.

The prime factorization method is not applicable directly for negative numbers, but we can find the prime factorization of 104 first.

Step 1: Finding the prime factors of 104: 104 = 2 × 2 × 2 × 13.

Step 2: The prime factors of 104 are 2³ × 13.

Step 3: The square root of 104 in radical form is simplified as √(2³ × 13).

Step 4: The square root of -104 is then √(2³ × 13) × i.

The long division method is used for finding the square root of non-perfect square positive numbers. For negative numbers, we first find the square root of the absolute value and then multiply by i.

Step 1: Apply the long division method to find √104.

Step 2: Group the digits of 104 as (1)(04).

Step 3: Find the largest number n whose square is ≤1. It is 1.

Step 4: Subtract 1² from 1 and bring down 04, making it 104.

Step 5: Double the divisor and find the next digit.

Step 6: Continue the process to find √104 ≈ 10.198. Step 7: Multiply by i to get √-104 ≈ 10.198 × i.

The approximation method involves estimating the square root by finding nearby perfect squares.

Step 1: Identify the closest perfect squares to 104, which are 100 and 121.

Step 2: √100 = 10, and √121 = 11, so √104 is between 10 and 11.

Step 3: Estimate √104 ≈ 10.198.

Step 4: Multiply by i for the square root of -104: √-104 ≈ 10.198 × i.

• Complex Number: A number comprising a real part and an imaginary part, often written as a + bi.
   
• Imaginary Unit: Denoted as i, it represents √-1 and is used to express complex numbers.
   
• Absolute Value: The non-negative value of a number without regard to its sign, applicable to both real and complex numbers.
   
• Modulus: The magnitude or absolute value of a complex number, calculated as the square root of the sum of the squares of its real and imaginary parts.
   
• Imaginary Number: A number that can be written in the form of a real number multiplied by the imaginary unit i.