Problem 56 - Algebra and Trigonometry

Question

Evaluate the radical expression, and express the result in the form \(a+b i\) $$ \frac{\sqrt{-7} \sqrt{-49}}{\sqrt{28}} $$

Step-by-Step Solution

Answer

The expression simplifies to \(-\frac{7}{2} + 0i\).

Step 1: Simplify the radicals in the numerator

Start by evaluating the product of the radicals in the numerator. Since \( \sqrt{-7} \cdot \sqrt{-49} = \sqrt{(-7) imes (-49)} \), we have \( \sqrt{343} \). Additionally, remember that \( \sqrt{-1} = i \), so \( \sqrt{-7} = \sqrt{7} i \) and \( \sqrt{-49} = 7i \). Multiply these, \( (\sqrt{7} \cdot 7) i^2 = -7\sqrt{7} \). So, the numerator simplifies to \(-7 \sqrt{7}\).

Step 2: Simplify the denominator

Identify \( \sqrt{28} \). The number 28 can be factored into \( 4 \times 7 \), so \( \sqrt{28} = \sqrt{4 \times 7} = \sqrt{4} \cdot \sqrt{7} = 2\sqrt{7} \). The denominator simplifies to \( 2\sqrt{7} \).

Step 3: Divide the numerator by the denominator

Take the simplified numerator \(-7 \sqrt{7}\) and divide it by the simplified denominator \(2\sqrt{7}\). This gives \( \frac{-7\sqrt{7}}{2\sqrt{7}} \). Simplifying, since \( \sqrt{7} \) cancels out, the expression becomes \( -\frac{7}{2} \).

Step 4: Express the result in terms of a complex number

The division resulted in a purely real number \( -\frac{7}{2} \), which can be expressed in the form \( a + bi \) as \( -\frac{7}{2} + 0i \).

Key Concepts

Radical Expressions

Radical expressions involve roots, which are mathematical relationships indicating how many times a number must be multiplied by itself to achieve another number. The most common radical is the square root, represented as \( \sqrt{} \). In the given exercise, the goal is to evaluate a complex radical expression. We begin by simplifying radicals in the expression \( \frac{\sqrt{-7} \sqrt{-49}}{\sqrt{28}} \). When tackling radical expressions, remember to simplify them to a form that is easier to work with. For example:

• The square root of a negative number can be expressed using \( i \), since \( \sqrt{-1} = i \).
• Combine the radicands (numbers inside the square root) when possible to further simplify the expression.

In many cases, simplifying revolves around factoring numbers under the square root sign. It is important to break down numbers into their prime factors, as it makes simplification and ultimately solving the problem much more manageable.

Imaginary Unit

The imaginary unit, denoted by \( i \), is what allows us to work with square roots of negative numbers. The definition of \( i \) is \( \sqrt{-1} \). This is crucial because it helps simplify expressions like \( \sqrt{-7} \) and \( \sqrt{-49} \) in our problem. Here is how it applies:

• Since \( \sqrt{-7} = \sqrt{7} \cdot i \), the negative under the root is managed by the \( i \).
• Similarly, \( \sqrt{-49} = 7i \) because \( 49 \) is a perfect square but is negatively signed.

Understanding that \( i^2 = -1 \) is vital since it often appears when multiplying these expressions. It serves as the backbone of combining and simplifying complex expressions. Whenever you encounter a radical with a negative, think of \( i \) and use it to pivot the problem into more calculable territory.

Complex Form a + bi

Complex numbers are expressed in the form \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) represents the imaginary unit. In the exercise, the answer was eventually simplified to a real number \( -\frac{7}{2} \), expressed as \( -\frac{7}{2} + 0i \). This representation shows that complex numbers can be comprised entirely of real numbers with an imaginary part of zero:

• \( a \) is the real component, determining the position on the horizontal axis of the complex plane.
• \( bi \) is the imaginary component, indicating placement on the vertical axis.

Numbers that do not contain an imaginary part still fit into this complex number system, simply with their \( b \) value as 0. It’s a versatile framework, allowing a broader understanding and manipulation of expressions within the realm of mathematics.