Square Root of -84

The square root is the inverse of the square of a number. Since -84 is a negative number, its square root is not a real number. Instead, it can be expressed in terms of an imaginary number. The square root of -84 is expressed as √-84 = √(84) × i, where i is the imaginary unit with the property that i² = -1. Since 84 is not a perfect square, the square root of 84 is irrational and expressed as √84 = 2√21. Therefore, the square root of -84 is 2√21 × i.

Since -84 is a negative number, we need to express its square root using imaginary numbers. The methods for finding the square roots of positive numbers, such as prime factorization and long division, are applied to 84 instead.

The prime factorization method helps to express the square root in its simplest form. Let's break down 84 into its prime factors:

Step 1: Finding the prime factors of 84 Breaking it down, we get 2 x 2 x 3 x 7 = 2² x 3 x 7

Step 2: We take the square root of the positive part: √84 = √(2² x 3 x 7) = 2√21

Step 3: Including the imaginary unit, the square root of -84 is: √-84 = 2√21 × i

The long division method is used to find the square root of positive numbers. For -84, we focus on the square root of 84, and then include the imaginary unit. Here is how to find √84 using the long division method:

Step 1: Pair the digits of 84 from right to left, getting 84.

Step 2: Find the largest number whose square is less than or equal to 84. This number is 9, since 9² = 81.

Step 3: Subtract 81 from 84, which results in 3.

Step 4: Bring down zeros in pairs, making the new dividend 300.

Step 5: Double the current quotient (9) to get 18.

Step 6: Find a digit 'x' such that 18x × x is less than or equal to 300. The digit is 1, since 181 × 1 = 181.

Step 7: Subtract 181 from 300 to get the remainder of 119.

Step 8: Continue the process to get a more accurate decimal value for √84.

Step 9: The result for √84 is approximately 9.165. Thus, the square root of -84 is approximately 9.165 × i.

The approximation method finds the square root by estimating. For -84, we work with √84 and use an imaginary unit.

Step 1: Identify perfect squares around 84. The closest are 81 and 100.

Step 2: 84 is closer to 81, so we start with 9.

Step 3: Using interpolation, (84 - 81) / (100 - 81) = 3/19 ≈ 0.158

Step 4: Add this to 9 to get approximately 9.158. Therefore, the square root of -84 is approximately 9.158 × i.

• Imaginary Number: A number that, when squared, gives a negative result. It is expressed as 'i' where i² = -1.
   
• Complex Number: A number that combines a real number and an imaginary number in the form a + bi, where a and b are real numbers.
   
• Perfect Square: A number that is the square of an integer. For example, 81 is a perfect square because it is 9².
   
• Irrational Number: A number that cannot be expressed as a simple fraction. Its decimal goes on forever without repeating.
   
• Square Root: The original number that was squared to produce a given square. For example, the square root of 25 is 5.