Square Root of -128

The square root is the inverse of the square of the number. Since -128 is a negative number, it does not have a real square root. Instead, the square root of -128 is expressed in terms of imaginary numbers. In radical form, it is expressed as √-128 = √128 * i, where i is the imaginary unit. In exponential form, it can be written as (128)^(1/2) * i. Simplifying further, √128 = 8√2, so √-128 = 8√2 * i, which is an imaginary number.

Since -128 is a negative number, we use the concept of imaginary numbers to find its square root. The prime factorization method, long division method, and approximation method are not applicable for negative numbers in the context of real numbers. Instead, we find the square root of the absolute value and multiply it by i:

Step 1: Find the square root of the absolute value, 128.

Step 2: Include the imaginary unit i to represent the square root of -128.

To find the square root of 128, we use the prime factorization method.

Step 1: Find the prime factors of 128. Breaking it down, we get 2 x 2 x 2 x 2 x 2 x 2 x 2 = 2^7.

Step 2: Pair the prime factors. Since 128 is not a perfect square, we can pair the 2s: 2^7 = (2^3 * 2^3) * 2 = (8 * 8) * 2.

Step 3: The square root of 128 is 8√2. Therefore, the square root of -128 is 8√2 * i.

To find the square root of a negative number like -128, we use the imaginary unit i, where i^2 = -1.

Step 1: Find the square root of the positive number 128, which is 8√2.

Step 2: Multiply the result by i to account for the negative sign. So the square root of -128 is 8√2 * i.

The concept of negative square roots involves imaginary numbers, as real numbers do not have square roots for negative values.

Step 1: Recognize that for any negative number, the square root involves i.

Step 2: Calculate the square root of the positive equivalent, and multiply by i. For example, √-128 = √128 * i = 8√2 * i.

• Square root: The square root is the inverse operation of squaring a number. For negative numbers, it involves imaginary numbers.
   
• Imaginary number: An imaginary number is a number that can be expressed in the form of a real number multiplied by the imaginary unit i.
   
• Imaginary unit: The imaginary unit i is defined such that i^2 = -1, used to express square roots of negative numbers.
   
• Complex number: A complex number consists of a real part and an imaginary part, expressed as a + bi.
   
• Absolute value: The absolute value of a number is its non-negative value, used when finding square roots of negative numbers.