Square Root of -224

The square root is the inverse of the square of the number. Since -224 is a negative number, its square root involves an imaginary unit, i. The square root of -224 is expressed in both radical and exponential form. In radical form, it is expressed as √-224, whereas in exponential form, it is (-224)^(1/2). To express it in terms of real and imaginary numbers, we write √-224 = √224 * i, where √224 ≈ 14.9666. Hence, the square root of -224 is approximately 14.9666i, indicating that it is an imaginary number.

The prime factorization method is used for perfect square numbers. However, the square root of negative numbers involves imaginary numbers. Therefore, we use the multiplication of the square root of the positive part with the imaginary unit, i. Let's explore the following methods:

• Prime factorization method (for positive components)
• Imaginary unit multiplication

The prime factorization of a number involves expressing it as a product of prime numbers. Now let us look at how 224 is broken down into its prime factors:

Step 1: Finding the prime factors of 224 Breaking it down, we get 2 x 2 x 2 x 2 x 2 x 7: 2^5 x 7

Step 2: Now, we found the prime factors of 224. Since -224 is negative, we use the imaginary unit, i.

Thus, √-224 = √224 * i = 2^2.5 * √7 * i.

The imaginary unit multiplication method is used to find the square root of negative numbers. Let us now learn how to find the square root of -224 using this method:

Step 1: First, find the square root of the positive part of -224.

Step 2: The square root of 224 is approximately 14.9666.

Step 3: Multiply the result by the imaginary unit, i, to get √-224 = 14.9666i.

The approximation method can also be extended to include imaginary numbers when dealing with negative square roots. Let us learn how to approximate the square root of -224:

Step 1: Determine the closest perfect squares to 224. The closest are 196 (14^2) and 225 (15^2), so √224 is between 14 and 15.

Step 2: Approximate √224 as 14.9666 and multiply by i to find the square root of -224: 14.9666i.

• Square root: The square root of a number is a value that, when multiplied by itself, gives the original number. For negative numbers, it involves the imaginary unit, i.

• Imaginary unit (i): The imaginary unit i is defined as the square root of -1, allowing the representation of square roots of negative numbers.

• Complex number: A number composed of a real and an imaginary component, expressed as a + bi.

• Prime factorization: The process of breaking down a number into its smallest prime factors.

• Approximation: Estimating a value close to the actual value, often used in finding the square roots of non-perfect squares.