Problem 45 - Intermediate Algebra

Question

Write each of the following in terms of \(i\), perform the indicated operations, and simplify. $$ \sqrt{-3} \sqrt{-5} $$

Step-by-Step Solution

Answer

The expression simplifies to \(-\sqrt{15}\).

Step 1: Express Square Roots in Terms of i

The imaginary unit is defined as \(i = \sqrt{-1}\). Therefore, whenever we encounter a square root of a negative number, we can express it in terms of \(i\). For \(\sqrt{-3}\), write it as \(\sqrt{3}\times i\). Similarly, \(\sqrt{-5}\) can be expressed as \(\sqrt{5}\times i\). Thus:\[\sqrt{-3} = \sqrt{3} \cdot i\]\[\sqrt{-5} = \sqrt{5} \cdot i\]

Step 2: Multiply the Expressions

Use the expressions from Step 1 to perform the multiplication:\[(\sqrt{3} \cdot i) \times (\sqrt{5} \cdot i)\]Apply the associative property to group the terms:\[(\sqrt{3} \cdot \sqrt{5}) \cdot (i \cdot i)\]Simplify the square roots as a single square root by multiplying the numbers inside:\[\sqrt{15} \cdot i^2\]

Step 3: Simplify i^2

Recall that \(i^2 = -1\). Substitute \(-1\) for \(i^2\) in the expression derived in Step 2:\[\sqrt{15} \cdot (-1) = -\sqrt{15}\]

Step 4: Final Answer

The expression simplifies to the final result by taking into account all previous steps:\[-\sqrt{15}\]

Key Concepts

Imaginary Unit

The world of mathematics becomes broader when we introduce the imaginary unit, denoted as \(i\). This unit is defined by the equation \(i = \sqrt{-1}\). It offers a way to express square roots of negative numbers, something not possible within the real number system. Instead of attempting to find a real number whose square is negative, we make use of \(i\). For example, \(\sqrt{-5}\) can be rewritten as \(\sqrt{5} \times i\), indicating that it's the product of a real number square root and the imaginary unit. This allows mathematicians and engineers to work within a broader number system called the complex numbers, where real and imaginary components are used together. The ability to express negative square roots using \(i\) is a foundational pillar in the study of complex numbers.

Key points about the imaginary unit:\

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• The imaginary unit is represented as \(i\), where \(i = \sqrt{-1}\).
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• It allows the expression of square roots of negative numbers in terms of \(i\).
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• This extends real numbers to complex numbers, with a format \(a + bi\), where \(a\) and \(b\) are real numbers.
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Simplification of Expressions

Simplifying expressions involving complex numbers often requires careful manipulation and the use of algebraic properties. When working with an expression like \(\sqrt{-3} \sqrt{-5}\), the first step is to express each term using the imaginary unit. This transforms the expression to \((\sqrt{3} \cdot i)(\sqrt{5} \cdot i)\). When multiplying, remember that you can group terms conveniently, following algebraic principles.

The multiplication \((\sqrt{3} \cdot \sqrt{5})(i \cdot i)\) involves applying the associative property, which lets us combine \(\sqrt{3} \) and \(\sqrt{5}\) into \(\sqrt{15}\). Meanwhile, \(i \cdot i\) becomes \(i^2\). At this stage, it’s crucial to recall that \(i^2 = -1\). This understanding enables us to further simplify the expression to \(\sqrt{15} \cdot (-1) = -\sqrt{15}\). Therefore, the original complex multiplication simplifies to a real number expression.

Steps to simplify complex expressions:\

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• Convert each square root of a negative number into a product with \(i\).
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• Use algebraic properties to rearrange and group terms effectively.
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• Simplify \(i^2\) as \(-1\) to achieve a simplified expression.
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Properties of Square Roots

Understanding square roots in the context of complex numbers involves recognizing crucial properties that have both real and imaginary applications. Initially, square roots are designed to express values that, when multiplied by themselves, yield the original number. However, with negative numbers, we turn to properties involving the imaginary unit \(i\).

Consider the square root property \(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\), which remains relevant even within complex numbers. This property guides the manipulation of expressions such as \(\sqrt{-3} \cdot \sqrt{-5}\), allowing us to express it as \(\sqrt{15} \cdot i^2\). When working with negative numbers under the square root, though, it’s important to split and handle them differently by expressing them in terms of \(i\).

Key considerations for properties of square roots:\

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• Square root properties can be used to simplify expressions even with complex numbers, but always check the context of negative numbers.
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• Remember to convert negative square roots into expressions involving \(i\) for correct manipulation.
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• The properties of square roots, when correctly applied, enable a smooth transition from real number operations to complex number operations.
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