Problem 49 - Precalculus Mathematics for Calculus

Question

Evaluate the radical expression and express the result in the form \(a+b i\) $$\sqrt{-3} \sqrt{-12}$$

Step-by-Step Solution

Answer

The expression simplifies to \(-6\).

Step 1: Simplify each radical separately

We start by simplifying each individual radical expression separately. First, note that \( \sqrt{-3} \) can be rewritten using the imaginary unit \( i \): \( \sqrt{-3} = \sqrt{3} \cdot i \). Similarly, \( \sqrt{-12} \) can be written as \( \sqrt{12} \cdot i \). To simplify \( \sqrt{12} \), note that \( 12 = 4 \times 3 \), so \( \sqrt{12} = \sqrt{4} \times \sqrt{3} = 2 \sqrt{3} \). Thus, \( \sqrt{-12} = 2 \sqrt{3} \cdot i \).

Step 2: Multiply the simplified expressions

Next, multiply the simplified expressions obtained from Step 1: \( \sqrt{-3} \cdot \sqrt{-12} = (\sqrt{3} \cdot i) \cdot (2 \sqrt{3} \cdot i) \).

Step 3: Apply properties of multiplication and combine terms

Use the properties of multiplication to combine the terms: \( \sqrt{3} \cdot 2 \sqrt{3} = 2(\sqrt{3})^2 = 2 \times 3 = 6 \). Now, \( i \cdot i = i^2 = -1 \). So, the full multiplication is \( 6 \cdot -1 = -6 \).

Step 4: Express the final result in the form \(a + bi\)

The result of the multiplication is \( -6 \), which can be expressed in the form \( a + bi \) as \( -6 + 0i \), where \( a = -6 \) and \( b = 0 \).

Key Concepts

Complex Numbers

Complex numbers are a fascinating area of mathematics that extends the idea of real numbers. They are built on the foundation of real and imaginary parts.

• A complex number is expressed in the form \(a + bi\), where:
• \(a\) is the real part.
• \(b\) is the imaginary part.
• \(i\) is the imaginary unit, satisfying \(i^2 = -1\).

Combining real numbers with an imaginary unit allows for more comprehensive solutions to many mathematical problems. It's especially useful in engineering and physics. Complex numbers enable us to perform arithmetic operations and solve equations that involve roots of negative numbers. They also add to the elegance and depth of mathematical concepts, solving problems that otherwise have no solution in the realm of real numbers.

Imaginary Unit

The concept of the imaginary unit is central to understanding complex numbers. The imaginary unit, represented by \(i\), was invented to allow for the square root of negative numbers.

• The defining property of \(i\) is that \(i^2 = -1\).
• This means \(\sqrt{-1} = i\).
• Using \(i\), radicals involving negative numbers can be expressed as imaginary numbers.

Consider evaluating \(\sqrt{-3}\). Without \(i\), there would be no solution in the real number system. However, with \(i\), this becomes feasible: \(\sqrt{-3} = i \sqrt{3}\). The imaginary unit allows us to express solutions in a more meaningful way, often leading to new insights and applications, particularly in signal processing and quantum mechanics.

Simplifying Radicals

Simplifying radicals is a crucial step when dealing with expressions involving square roots. Often, simplification involves rewriting the radicals to make them more manageable.

• Start by breaking down the number inside the radical into its prime factors.
• For example, for \(\sqrt{12}\), since \(12 = 4 \times 3\), it can be rewritten as \(\sqrt{4} \times \sqrt{3}\).
• Since \(\sqrt{4} = 2\), this simplifies to \(2\sqrt{3}\).

When radicals involve negative numbers, use \(i\) to help express them. For example, \(\sqrt{-12}\) becomes \(2\sqrt{3} \cdot i\).Simplification helps in performing further operations such as multiplication or addition, making the expressions easier to handle. It is essential when combining or multiplying radicals, as it sets the foundation for reducing complex expressions into simpler forms.