Problem 45 - Algebra for College Students

Question

Write each of the following in terms of \(i\), perform the indicated operations, and simplify. For example, $$ \begin{aligned} \sqrt{-3} \sqrt{-8} &=(i \sqrt{3})(i \sqrt{8}) \\ &=i^{2} \sqrt{24} \\ &=(-1) \sqrt{4} \sqrt{6} \\ &=-2 \sqrt{6} \end{aligned} $$ $$\sqrt{-3} \sqrt{-5}$$

Step-by-Step Solution

Answer

The simplified form is \(-\sqrt{15}\).

Step 1: Express each square root in terms of i

First, separate the negative sign from each square root since the square root of a negative number can be expressed as a product involving the imaginary unit \(i\). This means \(\sqrt{-3} = i\sqrt{3}\) and \(\sqrt{-5} = i\sqrt{5}\).

Step 2: Multiply the expressions

Multiply the expressions obtained in Step 1: \(\sqrt{-3} \times \sqrt{-5} = (i\sqrt{3}) \times (i\sqrt{5})\).

Step 3: Simplify using properties of i

In this step, combine and simplify the terms: \( i \times i = i^{2} \) and \( i^{2} = -1 \). Thus, \( (i \sqrt{3})(i \sqrt{5}) = i^{2} \times \sqrt{15} = -1 \times \sqrt{15} = -\sqrt{15}\).

Step 4: Final Simplification

The multiplication results in a simpler form, where the expression becomes \(-\sqrt{15}\), indicating the operations are complete.

Key Concepts

Imaginary Unit

The imaginary unit, commonly represented as \(i\), is a fundamental concept used to deal with square roots of negative numbers. In mathematics, the imaginary unit is defined as \(i = \sqrt{-1}\). This fascinating concept allows us to extend the real number system to include complex numbers.

• Anything multiplied by \(i\) shifts it to the imaginary number space.
• One key property is that \(i^2 = -1\).

When working with problems involving square roots of negative numbers, \(i\) helps express these roots neatly. For instance, \(\sqrt{-a} = i\sqrt{a}\), where \(a\) is a positive real number. This approach ensures that calculations stay manageable and correct when dealing with negative radicands.
In solving complex expressions, knowing the properties and behavior of the imaginary unit streamlines simplification and expression into simpler forms.

Square Roots

Square roots are a fundamental operation in mathematics, and dealing with negative numbers under the square root brings complexity. The traditional square root function, \(\sqrt{x}\), is defined for non-negative numbers. However, when faced with negative numbers under the square root, the expression becomes undefined in the realm of real numbers.
To resolve this, mathematicians use the concept of the imaginary unit \(i\). By expressing negative square roots in terms of \(i\), such as \(\sqrt{-a} = i\sqrt{a}\), they become manageable. In practice, each square root of a negative number is split, extracting the negative. For example:

• \(\sqrt{-3} = i\sqrt{3}\)
• \(\sqrt{-5} = i\sqrt{5}\)

By turning square roots of negatives into a product with \(i\), we can continue to manipulate these expressions using familiar algebraic rules. This method is key in simplifying complex expressions and facilitates the combination and multiplication of negative square roots efficiently.

Simplification Techniques

Simplification techniques are essential when handling expressions involving complex numbers, especially when the expressions contain square roots of negative numbers. After representing negative square roots with the imaginary unit \(i\), the goal is often to simplify the expression thoroughly.

• Multiply like terms: When expressions involve \(i\), for instance \((i\sqrt{a})(i\sqrt{b})\), you first multiply like terms: \(i \cdot i = i^2\), which simplifies to \(-1\).
• Combine the roots: The real components inside the square roots multiply together as usual, such as \(\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}\).

The final expression often results in a more straightforward form by reducing and eliminating unnecessary complexity. For example, from \((i\sqrt{3})(i\sqrt{5})\) to the final result of \(-\sqrt{15}\) in the exercise above. Leveraging these simplification rules ensures expressions are as concise as possible, making further computations or interpretations easier.