Square Root of -120

The square root is the inverse of the square of the number. Since -120 is negative, it does not have a real square root. The square root of a negative number is an imaginary number. In the complex number system, the square root of -120 can be expressed as √(-120) = √(120) × i, where i is the imaginary unit, equal to the square root of -1. Therefore, the square root of -120 is an imaginary number.

To find the square root of a negative number like -120, we must use the concept of imaginary numbers. The square root of -120 can be represented in terms of √120 and i. Let's explore the methods used for finding the square root of the positive part:

• Approximation method
• Prime factorization method

The approximation method is useful for finding square roots of non-perfect squares. Here, we will approximate the square root of 120 and then include the imaginary unit.

Step 1: Identify the closest perfect squares to 120. The perfect squares nearest to 120 are 100 and 121. Thus, √120 is between 10 and 11.

Step 2: Estimate the decimal value using the formula: (Given number - smaller perfect square) ÷ (larger perfect square - smaller perfect square) (120 - 100) ÷ (121 - 100) = 20 ÷ 21 ≈ 0.95

Step 3: Add this decimal to the smaller integer root: 10 + 0.95 = 10.95

Therefore, the square root of 120 is approximately 10.95, and the square root of -120 is approximately 10.95i.

Prime factorization is another method for finding square roots, mainly for perfect squares. Though 120 is not a perfect square, we can factor it for better understanding.

Step 1: Factor 120 into prime factors: 120 = 2 × 2 × 2 × 3 × 5 = 2^3 × 3 × 5

Step 2: Pair the factors: Since 120 is not a perfect square, it cannot be paired completely, but we can find the square root of 120 as: √120 = √(2^2 × 3 × 5 × 2) = 2√(30)

Thus, the square root of -120 is expressed in terms of i: √(-120) = 2√(30) × i

Imaginary numbers play a crucial role in square roots of negative numbers. The imaginary unit i is defined as √(-1). Therefore, any square root of a negative number can be expressed using i.

For example, √(-120) = 2√(30) i.

• Imaginary Number: A number that can be written as a real number multiplied by the imaginary unit i, where i is the square root of -1.
   
• Complex Number: A number composed of a real part and an imaginary part, expressed as a + bi.
   
• Square Root: The value that, when multiplied by itself, gives the original number. For negative numbers, it involves the imaginary unit.
   
• Prime Factorization: The process of breaking down a number into its smallest prime factors.
   
• Approximation Method: A method to estimate the square root of non-perfect squares by identifying the closest perfect squares and using a formula to find a decimal approximation.