Problem 34 - Algebra 2 and Trigonometry

Question

In \(19-34,\) write each sum or difference in terms of \(i\) $$ \sqrt{0.2025}+\sqrt{-0.09} $$

Step-by-Step Solution

Answer

\(0.45 + 0.3i\)

Step 1: Simplify the Positive Square Root

Calculate the square root of 0.2025 without considering the imaginary unit. The square root of 0.2025 is 0.45 because \((0.45)^2 = 0.2025\). So, \(\sqrt{0.2025} = 0.45\).

Step 2: Simplify the Negative Square Root

Evaluate the square root of -0.09 by recognizing that it involves the imaginary unit. The square root of positive 0.09 is 0.3, so \(\sqrt{-0.09} = 0.3i\). This simplifies the negative square root in terms of \(i\).

Step 3: Combine Results in Terms of i

Add the results from Step 1 and Step 2. The expression becomes \(0.45 + 0.3i\). This is the sum expressed in terms of the imaginary unit \(i\).

Key Concepts

Imaginary Unit

In mathematics, the imaginary unit is denoted by the symbol \(i\). It represents the square root of -1. This might sound strange at first because we usually say there is no real number whose square is negative. The imaginary unit is a tool used to extend the real number system to include solutions to equations like \(x^2 = -1\), which don't have solutions using only real numbers. With \(i\), the equation \(x^2 = -1\) can be solved, as \(i^2 = -1\).

The imaginary unit allows us to work with 'imaginary numbers' and, when combined with real numbers, forms complex numbers. For example, if we have the number 4 and combine it with the imaginary number 3i, we get the complex number \(4 + 3i\). This is very useful in many areas of mathematics and engineering. Complex numbers are crucial when dealing with certain types of equations and systems.

Square Roots

A square root of a number is a value that, when multiplied by itself, gives that number. For example, the square root of 16 is 4 because \(4 \times 4 = 16\). The square root symbol is \(\sqrt{}\). In real numbers, only positive numbers have real square roots. However, when we encounter a negative number under the square root, we need to include the imaginary unit \(i\).

To find the square root of a negative number, like -0.09 in this case, you first find the square root of the positive part (0.09, which is 0.3) and then multiply by \(i\). Therefore, \(\sqrt{-0.09} = 0.3i\). The presence of \(i\) indicates that it's an imaginary number. Understanding square roots helps in simplifying complex expressions that include both real and imaginary numbers.

Simplifying Expressions

Simplifying expressions is about combining and rewriting mathematical expressions in a more compact, easier-to-understand form. With complex numbers, which include both real and imaginary parts, simplification often involves combining like terms.

In the given problem, we first find the square root of 0.2025 to be 0.45, which is a real number. Next, we take care of the negative square root \(\sqrt{-0.09}\), which becomes \(0.3i\) due to the imaginary unit. When we combine these results, the expression \(\sqrt{0.2025} + \sqrt{-0.09}\) is simplified to \(0.45 + 0.3i\).

This simplification involves understanding both real and imaginary components and ensuring each component is correctly expressed. Always check your work by ensuring each term is in its simplest form, as this makes further calculations more manageable and reduces the potential for errors.