91Ó°ÊÓ

Understanding the i-form expression is fundamental when dealing with complex numbers. In algebra, i is used to represent the imaginary unit, which is the square root of -1. By definition, i satisfies the equation i² = -1. When asked to write a number in i-form, particularly when the radicand (the number under the square root) is negative, we utilize i to express this value as a complex number.

For example, to convert \( \sqrt{-45} \) to i-form, we start by splitting it into \( \sqrt{45} \) and \( \sqrt{-1} \) and then rewrite \( \sqrt{-1} \) as i. This results in \( \sqrt{45} * i \) or, as we'll see after simplification, \( 3\sqrt{5}i \) which is the simplified i-form expression.

A key step in managing expressions with complex numbers is radicand simplification. This technique involves breaking down the number inside the square root to its simplest form. To simplify a square root (or a radicand), we look for perfect square factors since the square root of a perfect square is an integer.

Take the example of \( \sqrt{45} \). When simplifying, we identify that 45 is not a perfect square, but it does contain a perfect square factor, 9 (which is 3²). Therefore, we can rewrite \( \sqrt{45} \) as \( \sqrt{3^2 \times 5} \) and further to \( 3\sqrt{5} \), pulling out the perfect square as an integer. This process helps in facilitating operations with complex numbers, especially when expressing them in i-form.

When simplifying radicals, as in the case of the radicand simplification, prime factorization is a useful tool. It involves expressing a number as the product of its prime factors, which are the building blocks of all numbers. Primes are numbers greater than 1 that have no divisors other than 1 and themselves.

In our exercise, we're dealing with the radicand 45, which can be factored into 3 * 3 * 5. These are all prime numbers. The process of prime factorization aids us in identifying groups of equal factors, particularly groups of two for square root simplification. By extracting these groups from under the square root as single numbers, we simplify the expression fully. This step is key in the overall simplification, leading to the correct i-form expression.