Problem 49 - Algebra and Trigonometry

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 evaluates to \(-6 + 0i\).

Step 1: Simplify Each Radical

To simplify each radical expression, we first express the negative numbers inside the square roots using imaginary numbers. Recall that \(i = \sqrt{-1}\). Thus,\(\sqrt{-3} = \sqrt{3} \cdot i\) and \(\sqrt{-12} = \sqrt{12} \cdot i\).

Step 2: Multiply the Radicals

Next, use the property \(\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}\) to multiply the two radicals:\[ \sqrt{-3} \cdot \sqrt{-12} = (\sqrt{3} \cdot i) \cdot (\sqrt{12} \cdot i) = i^2 \cdot \sqrt{3 \cdot 12} = -1 \cdot \sqrt{36} \] Here, \(i^2 = -1\) and \(3 \cdot 12 = 36\).

Step 3: Evaluate the Expression

Compute \(\sqrt{36}\):\[ \sqrt{36} = 6 \]Thus, the expression becomes:\[ -1 \cdot 6 = -6 \]

Step 4: Express in Standard Form

The result \(-6\) is a real number which can be expressed as \(a + bi\) by letting \(a = -6\) and \(b = 0\). Therefore, the expression in the form \(a + bi\) is:\[ -6 + 0i \]

Key Concepts

Imaginary Numbers

Imaginary numbers arise when we attempt to take the square root of a negative number, which isn't possible within the set of real numbers. In mathematics, this is addressed by introducing the unit 'i', defined as the square root of -1. Therefore, the square of 'i' is \[ i^2 = -1 \]. This allows us to express square roots of negative numbers as products involving 'i'. For instance, to find \( \sqrt{-3} \), we represent it as \( \sqrt{3} \times i \), meaning we first take the square root of the positive part (3) and then multiply by 'i'.
This representation simplifies our computations and enables us to handle complex expressions involving both real and imaginary components with ease. Remember, imaginary numbers are combined with real numbers to form complex numbers, which are expressed in the form \(a + bi\). In this notation, 'a' is the real part, and 'b' is the imaginary part. Understanding how to manipulate these forms is crucial for mastery in handling complex mathematical problems.

Radical Expressions

A radical expression involves roots, often square roots, of numbers. Simplifying radical expressions is a key skill in mathematics, especially when dealing with numbers under a radical sign (i.e., square root). When negatives are found inside the square root, we express them using imaginary numbers like in the exercise. For instance, \( \sqrt{-12} \) becomes \( \sqrt{12} \times i \).
It's important to note when simplifying:

• Simplify under the square root by factoring the number into prime factors.
• Express the square root of negative numbers by isolating 'i'.

These principles help break down complicated expressions in manageable steps. Once simplified, they allow us to multiply and combine such expressions easily, facilitating further mathematical operations.

Multiplication of Radicals

The multiplication of radicals involves using the property \( \sqrt{a} \cdot \sqrt{b} = \sqrt{ab} \). This property helps in combining radical expressions into a single radical. It essentially reduces the complexity by combining what's inside the radicals.
In our exercise, when multiplying \( \sqrt{-3} \) and \( \sqrt{-12} \), we first simplify each using imaginary numbers. Then, by applying the multiplication property of radicals, \( \sqrt{3 \cdot 12} = \sqrt{36} \), we find the product to be \(-6\), exploiting the identity \( i^2 = -1 \).
Understanding how to use these properties effectively allows us to resolve even complex radical expressions efficiently and accurately. By following structured steps, these expressions are transformed into standard forms, such as \( a + bi \), making them easier to interpret and use within a greater range of mathematical contexts.