Problem 55 - Intermediate Algebra

Question

Write each of the following in terms of \(i\), perform the indicated operations, and simplify. $$ \frac{\sqrt{-25}}{\sqrt{-4}} $$

Step-by-Step Solution

Answer

The simplified form is \(\frac{5}{2}\).

Step 1: Express in terms of i

To write the square roots of negative numbers in terms of i, we use the property that \(\sqrt{-a} = i\sqrt{a}\). Thus, \\[\sqrt{-25} = i\sqrt{25} = 5i \\sqrt{-4} = i\sqrt{4} = 2i\]

Step 2: Substitute back into the equation

Replace \(\sqrt{-25}\) and \(\sqrt{-4}\) with their equivalent expressions in terms of \(i\):\[ \frac{5i}{2i} \]

Step 3: Simplify the expression

Cancel out the \(i\) from the numerator and the denominator:\[ \frac{5i}{2i} = \frac{5}{2} \]Thus, the expression simplifies to \(\frac{5}{2}\).

Key Concepts

Imaginary Unit

Understanding the imaginary unit is essential to solving expressions involving square roots of negative numbers. The imaginary unit is denoted by the symbol \(i\). It is defined as the square root of \(-1\), which is a number that doesn't exist in the realm of real numbers. However, it provides a way to work with quadratic equations that do not have real solutions.

This concept allows us to handle the square roots of negative numbers. For example, \(i^2\) equals \(-1\), showing how \(i\) serves as a foundation for solving equations that include negative square roots. This unique property of the imaginary unit is what makes it possible to simplify and work with expressions that seem unsolvable in the real number world.

Simplifying Expressions

Simplifying expressions involving imaginary numbers often requires expressing terms with the imaginary unit \(i\) and then performing algebraic operations as one would with real numbers.

When simplifying an expression, like \( \frac{\sqrt{-25}}{\sqrt{-4}} \), the first step is to express the square roots of negative numbers in terms of \(i\). With our example, this becomes \( \frac{5i}{2i} \).

• First, express each square root in terms of \(i\).
• Substitute these back into the expression.
• Then, simplify by cancelling common terms, just as you would with regular fractions.

This approach leaves us with a simplified expression that is easier to understand and work with.

Square Roots of Negative Numbers

The square roots of negative numbers can be confusing, but they become manageable when you understand how to convert them using the imaginary unit \(i\). Normally, the square root of a positive number \(a\) is straightforward — it's simply a number \(b\) such that \(b^2 = a\). But negative numbers are trickier because no real number squared will give a negative result.

This is where \(i\) steps in. For any negative number \(-a\), we use \(\sqrt{-a} = i\sqrt{a}\). So, to process the square root of \(-25\), for example, you would express this as \(i\sqrt{25} = 5i\).

By expressing the square roots of negative numbers in this way, you can handle and simplify calculations involving them. This makes complex number operations both doable and understandable in many mathematical applications.