Problem 26 - Algebra 2 and Trigonometry

Question

In \(19-34,\) write each sum or difference in terms of \(i\) $$ \sqrt{50}+\sqrt{-2}+\sqrt{200}+\sqrt{-50} $$

Step-by-Step Solution

Answer

The sum or difference is \(15\sqrt{2} + 6i\sqrt{2}\).

Step 1: Simplify Each Square Root

Start by identifying and simplifying each of the square roots. The expression becomes:\(\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}\)\(\sqrt{-2} = \sqrt{2}i = i\sqrt{2}\)\(\sqrt{200} = \sqrt{100 \times 2} = 10\sqrt{2}\)\(\sqrt{-50} = \sqrt{50}i = 5i\sqrt{2}\).

Step 2: Combine Like Terms

After simplifying, we combine like terms. We see that - The real components in \(\sqrt{50}\) and \(\sqrt{200}\) can be combined as \(5\sqrt{2} + 10\sqrt{2} = 15\sqrt{2}\).- The imaginary components in \(\sqrt{-2}\) and \(\sqrt{-50}\) are combined as \(i\sqrt{2} + 5i\sqrt{2} = 6i\sqrt{2}\).

Step 3: Write the Final Expression

After combining the terms, the expression in terms of \(i\) is:\(15\sqrt{2} + 6i\sqrt{2}\).

Key Concepts

Understanding the Imaginary Unit

The imaginary unit is a core concept when dealing with complex numbers. It is denoted by the symbol \(i\), and it represents the square root of \(-1\). This means that \(i^2 = -1\). Since real numbers cannot be squared to produce a negative result, mathematicians introduced \(i\) to address this limitation in arithmetic.

• For example, \(\sqrt{-2}\) can be expressed as \(\sqrt{2} \times i = i\sqrt{2}\).
• Similarly, any negative number inside a square root can be transformed using the imaginary unit.

Understanding \(i\) is crucial when you encounter square roots of negative numbers, helping you to simplify complex expressions. It allows for broader solutions in mathematics beyond the realm of real numbers. By grasping the concept of \(i\), you can efficiently simplify expressions involving square roots of negative numbers, as seen in the original exercise.

Demystifying Square Roots

Square roots are numbers that, when multiplied by themselves, yield a given value. Expressed as \(\sqrt{x}\), they are a fundamental part of many mathematical expressions.

• The square root of a positive number is straightforward. For example, \(\sqrt{50}\) becomes \(\sqrt{25 \times 2} = 5\sqrt{2}\).
• However, the square root of a negative number requires the use of the imaginary unit \(i\). For instance, \(\sqrt{-50}\) becomes \(\sqrt{50}i = 5i\sqrt{2}\).

Breaking down the square roots into their simplest form involves identifying perfect squares within the radicand (the number under the square root sign). This method allows for easier calculations and simplification further in the problem-solving process.
In practice, simplifying square roots is an essential skill. It can make complex expressions more manageable and facilitate easier additions or subtractions of terms with square roots, as performed in the given exercise.

Simplifying Expressions Involving Square Roots and Imaginary Numbers

Simplifying mathematical expressions is about making them shorter and easier to handle while retaining their original value. In the context of the original problem, it involves combining terms efficiently.

• Let's start with like terms. Expressions like \(5\sqrt{2} + 10\sqrt{2}\) can be combined to \(15\sqrt{2}\) by the distributive property, which allows you to add their coefficients.
• Similarly, imaginary components such as \(i\sqrt{2} + 5i\sqrt{2}\) are combined to form \(6i\sqrt{2}\).

When simplifying expressions that include both real and imaginary parts, it's important to recognize and group them accordingly.
This practice ensures that the expression is as simplified as possible, making any further mathematical operations, such as solving equations or plotting, more straightforward. With practice, simplifying expressions becomes intuitive, aiding in problem-solving efficiency and accuracy.