Problem 50 - Algebra and Trigonometry

Question

Evaluate the radical expression, and express the result in the form \(a+b i\) $$ \sqrt{4} \sqrt{-27} $$

Step-by-Step Solution

Answer

The expression simplifies to \(0 + 6\sqrt{3} i\).

Step 1: Break Down the Expression

We start by evaluating each square root individually. The expression given is \( \sqrt{4} \sqrt{-27} \). First, find \( \sqrt{4} \) and \( \sqrt{-27} \) separately.

Step 2: Evaluate \(\sqrt{4}\)

The expression \( \sqrt{4} \) is straightforward. Since 4 is a perfect square, \( \sqrt{4} = 2 \).

Step 3: Simplify \(\sqrt{-27}\)

For \( \sqrt{-27} \), we rewrite it using the imaginary unit \( i \), where \( i = \sqrt{-1} \). Thus, \( \sqrt{-27} = \sqrt{27} \cdot \sqrt{-1} = \sqrt{27} \cdot i \).

Step 4: Simplify \(\sqrt{27}\)

The number 27 can be factored into \(3^3\), so \( \sqrt{27} = \sqrt{3^3} = \sqrt{3^2 \cdot 3} = 3\sqrt{3} \). Therefore, \( \sqrt{-27} = 3\sqrt{3} \cdot i \).

Step 5: Multiply the Results

Now multiply the two simplified results: \( 2 \cdot 3\sqrt{3} \cdot i = 6\sqrt{3} i \).

Step 6: Express the Result in \(a + bi\) Form

Since the expression \(6\sqrt{3} i\) is purely imaginary, it can be expressed in the form \(0 + 6\sqrt{3} i\).

Key Concepts

Imaginary Unit

In the realm of mathematics, the imaginary unit is a fundamental concept when dealing with complex numbers. It's symbolized by the letter \(i\) and is defined as \(i = \sqrt{-1}\). This definition allows us to work with the square roots of negative numbers, which are not possible in the realm of real numbers.

• The imaginary unit makes it easy to express complex numbers, which consist of a real part and an imaginary part, in the form \(a + bi\).


• When squared, the imaginary unit returns a real number: \(i^2 = -1\).

This property of imaginary numbers allows mathematicians and engineers to solve equations and model phenomena in ways that real numbers alone can't achieve. For example, in electrical engineering and physics, the imaginary unit is extensively used to describe oscillating waves.

Radical Expressions

Radical expressions involve roots, such as square roots or cube roots, and play a significant role in simplifying mathematical problems. A radical expression can include numbers, variables, and combinations thereof under a radical sign, also known as a root symbol (√).

• To simplify radical expressions, factors under the radical can be extracted as multiples of roots outside the radical.


• For example, \(\sqrt{27}\) can be rewritten as \(3\sqrt{3}\) since 27 is \(3^3\).

The process of manipulating these expressions often involves prime factorization and identifying perfect squares (or cubes, etc.). It's a crucial skill that helps in simplifying complex problems, which is beneficial in algebra, calculus, and many applied sciences.

Square Roots

Square roots are perhaps the most common type of root found in mathematics. They ask us to determine what value, when multiplied by itself, gives the number under the radical. For positive numbers, the square root is straightforward. However, square roots of negative numbers open up the domain of complex numbers.
Here are a few important points about square roots:

• The square root symbol, \(\sqrt{}\), typically implies the principal (positive) square root.


• For negative arguments under the square root, we use the imaginary unit \(i\) to represent them (e.g., \(\sqrt{-9} = 3i\)).

Understanding square roots is essential as they frequently appear in various forms of equations, from quadratic equations to geometry and beyond.

Simplifying Radicals

Simplifying radicals is a method used to make expressions involving roots more manageable. It is a systematic approach that can transform complex radical expressions into simpler equivalents. The goal is to express the number under the radical as a product of its prime factors and identify and extract perfect squares.
Let's walk through the process using \(\sqrt{27}\) as an example:

• Break down 27 into its prime factors: \(27 = 3^3\).


• Rewrite the expression: \(\sqrt{27} = \sqrt{3^3} = \sqrt{3^2 \times 3}\).


• Extract the perfect square: this gives us \(3\sqrt{3}\).

Once simplified, radicals are easier to manipulate, which is particularly helpful when performing operations such as addition, subtraction, multiplication, or solving equations involving radicals.