Square Root of -136

The square root is the inverse of the square of a number. In the case of negative numbers, the square root involves imaginary numbers. The square root of -136 is expressed in the form of √-136, which can be rewritten as √136 × i, where i is the imaginary unit. The numerical value of √136 is approximately 11.662. Therefore, the square root of -136 is 11.662i, an imaginary number.

The square root of a negative number involves the imaginary unit 'i'. Common methods to calculate square roots include factoring for perfect squares, long division for non-perfect squares, and approximation. However, for negative numbers, we first compute the square root of the positive counterpart and then multiply by i. Let's examine these methods:

• Prime factorization method
• Long division method
• Approximation method

Prime factorization involves breaking down a number into its prime components. While -136 is not a perfect square, we can find the prime factorization of 136:

Step 1: Finding the prime factors of 136 Breaking it down, we get 2 × 2 × 2 × 17: 2³ × 17

Step 2: Since -136 is negative, the square root involves 'i'. Thus, √-136 = √136 × i = 2√34 × i

The long division method is used for non-perfect squares. Let's use this method to find √136, then multiply by i.

Step 1: Group digits of 136 from right to left: 36 and 1.

Step 2: Find n such that n² ≤ 1. Here, n = 1 as 1 × 1 = 1.

Step 3: Subtract 1 from 1 to get 0, then bring down 36.

Step 4: Add 1 to itself to get 2 as the new divisor.

Step 5: Find 2n × n ≤ 36. Let n = 6, then 26 × 6 = 156; adjust to get 216.

Step 6: Subtract 216 from 360 to get 144.

Step 7: Add decimal and bring down two zeroes to get 14400.

Step 8: Find the new divisor. Continue the division until reaching a close approximation. Result: √136 ≈ 11.662, so √-136 = 11.662i

The approximation method is used for quick calculations. We find the square root of 136 and multiply by i.

Step 1: Identify the perfect squares around 136: 121 (11²) and 144 (12²).

Step 2: 136 is closer to 144, so √136 ≈ 11.662

Step 3: Therefore, the square root of -136 is 11.662i

• Imaginary Unit: The imaginary unit 'i' is defined as the square root of -1, used to express the square roots of negative numbers.
   
• Complex Number: A number that comprises a real part and an imaginary part, typically written in the form a + bi.
   
• Square Root: The square root of a number is a value that, when multiplied by itself, gives the original number. For negative numbers, involves the imaginary unit.
   
• Approximation: A method of finding an estimated value close to the actual value, used here for finding square roots of non-perfect squares.
   
• Prime Factorization: The process of expressing a number as the product of its prime factors, used in simplifying square roots.