Square Root of -288

The square root is the inverse of the square of the number. Since -288 is a negative number, its square root is not a real number. The square root of -288 is expressed in terms of the imaginary unit 'i'. In radical form, it is expressed as √(-288) = √288 × i, while in exponential form, it is (288)^(1/2) × i. The square root of 288 is 16.97056, so the square root of -288 is 16.97056i, which is an imaginary number.

The prime factorization method is used for perfect square numbers. However, since -288 is not a perfect square and is negative, we focus on finding the square root of 288 first, then multiply by 'i'. Let's learn the following methods:

• Prime factorization method
• Long division method
• Approximation method

The product of prime factors is the prime factorization of a number. Now, let us look at how 288 is broken down into its prime factors:

Step 1: Finding the prime factors of 288 Breaking it down, we get 2 × 2 × 2 × 2 × 3 × 3: \(2^4 \times 3^2\)

Step 2: Now, we found out the prime factors of 288. The second step is to make pairs of those prime factors. Since 288 is not a perfect square, the digits of the number can’t be grouped perfectly into pairs. Therefore, calculate √288 using prime factorization: \(2^2 \times 3\) yields 12√2.

Step 3: The square root of -288 is thus √288 × i = 12√2 × i.

The long division method is particularly used for non-perfect square numbers. In this method, we check the closest perfect square number for the given number and then multiply by 'i'. Let's learn how to find the square root step by step:

Step 1: Group the numbers from right to left for 288, which is grouped as 88 and 2.

Step 2: Find n whose square is 2. We use 1 because 1 × 1 is less than or equal to 2. The quotient is 1, and after subtracting 1 from 2, the remainder is 1.

Step 3: Bring down 88, making the new dividend 188. Add the old divisor with itself, 1 + 1, to get 2 as the new divisor.

Step 4: Find 2n × n ≤ 188. Let n be 7, because 27 × 7 = 189.

Step 5: Subtract 188 from 189 to get a remainder of 1.

Step 6: Since we have a remainder, we continue the division process. The quotient becomes 16.97056, so the square root of 288 is approximately 16.97056.

Step 7: Since we are dealing with -288, the square root of -288 is 16.97056i.

Approximation is another method for finding square roots, which is easy for estimating the square root of a given number, then multiplying by 'i'. Now, let us learn how to find the square root of -288 using approximation:

Step 1: Find the closest perfect square of √288. The closest perfect squares to 288 are 256 and 289. √288 falls between 16 and 17.

Step 2: Apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula (288 - 256) / (289 - 256) = 0.97.

Step 3: Add this decimal to the integer square root. 16 + 0.97 = 16.97. Step 4: The square root of -288 is 16.97i.

• Imaginary Number: A number that can be expressed as a real number multiplied by the imaginary unit 'i', where i² = -1.

• Square Root: The square root of a number is a value that, when multiplied by itself, gives the original number.

• Imaginary Unit (i): The imaginary unit 'i' is defined as √(-1), used to represent the square roots of negative numbers.

• Prime Factorization: The expression of a number as a product of its prime factors.

• Approximation Method: A method used to estimate the value of a square root for non-perfect squares.