Problem 20 - Algebra 2 and Trigonometry

Question

In \(19-34,\) write each sum or difference in terms of \(i\) $$ \sqrt{-25}-\sqrt{-4} $$

Step-by-Step Solution

Answer

The expression simplifies to \(3i\).

Step 1: Express Square Roots of Negative Numbers in Terms of 'i'

To convert square roots of negative numbers into expressions with 'i', recall that \(i\) is defined as \(\sqrt{-1}\). Therefore, \(\sqrt{-25} = \sqrt{25} \times \sqrt{-1} = 5i\), and \(\sqrt{-4} = \sqrt{4} \times \sqrt{-1} = 2i\).

Step 2: Write the Expression as a Difference Involving 'i'

Given that \(\sqrt{-25} = 5i\) and \(\sqrt{-4} = 2i\), substitute these values in the original expression \(\sqrt{-25} - \sqrt{-4}\). This results in \(5i - 2i\).

Step 3: Simplify the Expression

Combine the terms involving 'i': \(5i - 2i = 3i\). Thus, the expression simplifies to \(3i\).

Key Concepts

Understanding the Imaginary Unit

The imaginary unit, commonly represented as \(i\), is a fundamental concept in the study of complex numbers. It is defined as the square root of \(-1\). This concept allows us to extend the number system to include solutions for equations like \(x^2 + 1 = 0\), which do not have real solutions. When disrupting the traditional rules of numbers by introducing \(i\), mathematicians are able to explore roots of negative numbers and complex analysis more freely.

• Defined as \(i^2 = -1\).
• Crucial for expressing solutions to equations involving square roots of negative numbers.
• Used to express complex numbers of the form \(a + bi\), where \(a\) and \(b\) are real numbers.

These properties of \(i\) enable us to represent and work with numbers that incorporate imaginary components, thus broadening our understanding beyond real numbers.

Simplifying Expressions Involving 'i'

When simplifying expressions that include the imaginary unit \(i\), the process is relatively straightforward but requires diligence in separating real components from imaginary components. Here's how you can approach it:

• Convert negative square roots to expressions involving \(i\). For instance, \(\sqrt{-25}\) becomes \(5i\) because \(\sqrt{-25} = \sqrt{25} \times \sqrt{-1} = 5i\).
• Add or subtract imaginary terms directly as you would with like terms in algebra. For example, in the expression \(5i - 2i\), simply combine the \(i\) terms to get \(3i\).
• Always express the final result in simplest form, retaining the \(a + bi\) format when possible.

By keeping the steps ordered and clear, handling expressions with imaginary numbers becomes as systematic as with any algebraic expression.

Square Roots of Negative Numbers

Square roots of negative numbers can be puzzling at first, as traditional square root concepts don't naturally apply. But by using the imaginary unit \(i\), we can redefine and manipulate these expressions easily.

• For any negative number \(-a\), the square root \(\sqrt{-a}\) can be rewritten as \(\sqrt{a} \times i\). For instance, \(\sqrt{-4}\) becomes \(2i\), since \(\sqrt{4} = 2\) and \(\sqrt{-1} = i\).
• This transformation allows us to employ square roots in broader mathematical contexts, incorporating them into complex numbers.
• When these square roots are involved in expressions, like a sum or difference, they follow basic arithmetic rules. E.g., the expression \(\sqrt{-25} - \sqrt{-4} = 5i - 2i\) simplifies neatly to \(3i\).

By utilizing \(i\), the square roots of negative numbers can be consistently treated, greatly expanding their application in various mathematical fields.