Problem 14 - Algebra 2 and Trigonometry

Question

In \(3-18,\) write each number in terms of \(i\) $$ \sqrt{-500} $$

Step-by-Step Solution

Answer

\(\sqrt{-500} = 10 \sqrt{5} i\)

Step 1: Understand the Concept of Complex Numbers

Complex numbers involve the imaginary unit, denoted as \(i\), where \(i = \sqrt{-1}\). With this, any square root of a negative number can be expressed in terms of \(i\).

Step 2: Simplify the Negative Square Root

Since \(\sqrt{-500}\) involves a negative inside the square root, rewrite this as \(\sqrt{500} \times \sqrt{-1}\). The \(\sqrt{-1}\) part becomes \(i\), so we have \(\sqrt{500}\times i\).

Step 3: Simplify the Square Root of the Positive Part

Find the prime factorization of 500, which is \(500 = 5^3 \times 2^2\). We can pair factors for simplification: \(\sqrt{500} = \sqrt{(2^2 \times 5^2) \times 5} = \sqrt{(2 \times 5)^2} \times \sqrt{5}\). Thus, \(\sqrt{500} = 10 \times \sqrt{5}\).

Step 4: Combine All Parts

Substitute the simplified part back into the expression: \(\sqrt{-500} = 10 \sqrt{5} \times i\). Therefore, the expression is simplified to \(10 \sqrt{5} i\).

Key Concepts

Imaginary Unit

Complex numbers introduce us to the imaginary unit, represented by the letter \(i\). This imaginary unit is defined by the property \(i = \sqrt{-1}\). It's called "imaginary" not because it's fictitious, but because it allows for the expression of numbers that are not part of the real number line. When we encounter a square root with a negative number, like \(\sqrt{-4}\), we can express it as \(2i\), because \(\sqrt{-4} = \sqrt{4} \times \sqrt{-1} = 2 \times i\).
Understanding this concept is crucial when working with complex numbers, as it permits us to extend the idea of a number beyond the confines of the conventional real number system. The imaginary unit \(i\) is a fundamental building block in many areas of mathematics, engineering, and physics, opening doors to advanced topics in waves, electrical engineering, and more.

Square Root Simplification

Simplifying square roots is a fundamental skill in algebra, particularly when dealing with both real and complex numbers. To simplify expressions such as \(\sqrt{-500}\), we first need to consider the square root of the negative number separately.

• First, break it into the product of \(\sqrt{500}\) and \(\sqrt{-1}\).
• The \(\sqrt{-1}\) component will become \(i\), as previously described.
• Next, simplify \(\sqrt{500}\) by finding its factors.

Upon finding factors, identify perfect square factors for simplification. For instance, \(500\) can be decomposed into \((2^2 \times 5^2) \times 5\). The square root of \((2 \times 5)^2\) simplifies to \(10\), and \(\sqrt{5}\) remains under the root since 5 is not a perfect square.
Putting it together: \(\sqrt{500} = 10\sqrt{5}\), and thus \(\sqrt{-500} = 10\sqrt{5}\times i\). This process helps us handle complex roots with ease, translating them into forms usable in real-world problems.

Prime Factorization

Prime factorization is an essential process in mathematics used to break down numbers into their basic building blocks, which are prime numbers. A prime number is a number greater than 1 that can only be divided by 1 and itself without leaving a remainder.

In problems involving square roots, like \(\sqrt{500}\), prime factorization helps in simplifying the expression:

• First, identify the prime factors of 500.
• You can decompose 500 as \(2^2 \times 5^3\).
• Pair up the powers of prime numbers wherever possible.

In the case of 500, \(5^2\) and \(2^2\) are perfect squares, and can thus be taken out of the square root, simplifying it to \(10\sqrt{5}\).
At the root of many algebraic problems, prime factorization simplifies calculations and is a powerful tool in both simple and complex mathematical contexts. It's especially helpful when converting complex numbers into more usable forms.