Square Root of -98

The square root is the inverse operation of squaring a number. Since -98 is negative, it does not have a real number square root. The square root of -98 is expressed in terms of imaginary numbers. In radical form, it is expressed as √(-98), or equivalently as √98i, where i is the imaginary unit with the property that i² = -1. The value of √98 is approximately 9.899, so √(-98) = 9.899i.

Finding the square root of a negative number involves using the imaginary unit i. We express the square root of a negative number in terms of i and the positive part of the number. For √(-98), we first calculate the square root of 98, and then multiply by i:

1. Calculate √98.

2. Multiply the result by i.

The prime factorization method is used to simplify the square root of positive numbers. For negative numbers, we factor the positive part. Let’s see how 98 is broken down into its prime factors:

Step 1: Prime factorize 98. 98 = 2 × 7 × 7 = 2 × 7²

Step 2: Use the prime factors to simplify: √98 = √(2 × 7²) = 7√2

Step 3: Multiply by i: The square root of -98 is 7√2i.

The long division method is not applicable to negative numbers directly for real square roots, but it can be used to approximate the square root of the positive part (98) before multiplying by i.

Step 1: Group digits of 98 from right to left as 98.

Step 2: Find an n such that n² is closest to 98. Here, n = 9 since 9² = 81.

Step 3: The approximate square root of 98 is 9.899.

Step 4: Multiply by i: √(-98) = 9.899i.

The approximation method can also be used for the positive part of -98.

Step 1: Find the closest perfect squares around 98. The closest perfect squares to 98 are 81 (9²) and 100 (10²).

Step 2: Estimate √98 using these perfect squares. √98 is approximately 9.899.

Step 3: Multiply by i to get the square root of -98: √(-98) = 9.899i.

• Imaginary unit: The imaginary unit is denoted as i and represents √(-1). It is used to express the square roots of negative numbers.
   
• Complex number: A number comprising a real part and an imaginary part, expressed as a + bi, where a and b are real numbers.
   
• Real number: A value representing a quantity along a continuous line, including all rational and irrational numbers.
   
• Square root: The square root of a number x is a number y such that y² = x. For negative numbers, it involves the imaginary unit.
   
• Prime factorization: The process of expressing a number as the product of its prime factors, used to simplify square roots of positive numbers.