Square Root of -164

The square root is the inverse of the square of a number. Since -164 is a negative number, its square root involves imaginary numbers. The square root of -164 is expressed in both radical and exponential form. In the radical form, it is expressed as √(-164), whereas (-164)^(1/2) in the exponential form. The square root of a negative number is an imaginary number, which can be expressed as i√164, where i is the imaginary unit with the property that i² = -1.

To find the square root of a negative number like -164, we need to recognize that it involves the imaginary unit i. The prime factorization and long division methods do not apply directly to negative numbers since they yield real numbers. However, we can find the square root of the positive counterpart and then multiply it by i. Let's explore the steps:

1. Find the square root of 164 using the usual methods.

2. Multiply the result by i to account for the negative sign.

The product of prime factors is the prime factorization of a number. Here’s how 164 is broken down into its prime factors:

Step 1: Finding the prime factors of 164 Breaking it down, we get 2 x 2 x 41: 2² x 41

Step 2: Pair the prime factors. Since 164 is not a perfect square, the digits of the number can’t be grouped into pairs for complete pairs.

Thus, the square root of 164 is expressed as 2√41, and for -164 as i(2√41).

The long division method is typically used for finding the square roots of positive non-perfect square numbers. Here’s how to find the square root of 164 using the long division method, then apply the result to -164:

Step 1: Group the numbers from right to left. For 164, group as 64 and 1.

Step 2: Find n whose square is 1. Here, n is 1 because 1 x 1 = 1. Subtract 1 from 1, resulting in a remainder of 0.

Step 3: Bring down 64, the new dividend. Add the previous divisor 1 to itself to get 2, the new divisor.

Step 4: Find n such that 2n x n ≤ 64. Here, n is 8 because 28 x 8 = 224, and 224 is less than 640.

Step 5: Subtract 224 from 640 to get a remainder of 416 and continue the process to get more decimal places. Once you find the square root of 164, multiply the result by i for -164.

The approximation method is another way to find square roots. Here’s how to find the square root of 164 using the approximation method:

Step 1: Identify the closest perfect squares to 164. The nearest perfect squares are 144 (12²) and 169 (13²). Thus, √164 is between 12 and 13.

Step 2: Use the approximation formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Applying it: (164 - 144) / (169 - 144) = 20 / 25 = 0.8 Add the decimal to the lower bound: 12 + 0.8 = 12.8 For -164, multiply the result by i to get i(12.8).

• Imaginary Unit: The imaginary unit i is defined such that i² = -1. It is used to express the square roots of negative numbers.

• Square Root: The square root of a number x is a number y such that y² = x. For negative numbers, the square root involves the imaginary unit i.

• Prime Factorization: Breaking down a number into the product of its prime factors. For example, 164 = 2² x 41.

• Perfect Square: A number that is the square of an integer. For example, 144 is a perfect square since 12² = 144.

• Approximation: The process of finding a close estimate of a number, often used for non-perfect squares.