Problem 52 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The result is \(-\frac{7}{2} + 0i\).

Step 1: Simplify the radicands

Recognize that \(\sqrt{-7} = \sqrt{7}i\) and \(\sqrt{-49} = \sqrt{49}i\) because the square root of a negative number introduces the imaginary unit \(i\). Thus, \(\sqrt{-7} = \sqrt{7}i\) and \(\sqrt{-49} = 7i\).

Step 2: Simplify the product in the numerator

Calculate the product of the two imaginary radicals in the numerator: \(\sqrt{7}i \times 7i = 7 \times \sqrt{7} \times i^2\). Since \(i^2 = -1\), simplify to \(-7\sqrt{7}\).

Step 3: Simplify the denominator

Calculate the square root in the denominator: \(\sqrt{28} = \sqrt{4 \times 7} = \sqrt{4} \times \sqrt{7} = 2\sqrt{7}\).

Step 4: Divide the expression

Divide the result from step 2 by step 3: \(\frac{-7\sqrt{7}}{2\sqrt{7}}\). The \(\sqrt{7}\) terms cancel out, leaving \(-\frac{7}{2}\).

Step 5: Write the expression as a complex number

Since the result is purely real, write the expression in the form of a complex number: \(-\frac{7}{2} + 0i\).

Key Concepts

Imaginary Unit

The imaginary unit, denoted as \( i \), is a fundamental concept in the field of complex numbers. It is defined as \( i = \sqrt{-1} \). Breaking down why the imaginary unit is important, let’s consider the need to take square roots of negative numbers. In mathematics, the square root of any negative number would not be possible within the scope of only real numbers. This led to the concept of the imaginary unit.
- Whenever you see a negative number under a square root, the imaginary unit comes into play- For instance, \( \sqrt{-4} \) becomes \( \sqrt{4}i = 2i \) since \( \sqrt{-1} = i \)- It allows us to express numbers with both real parts and imaginary parts, forming complex numbers like \( a + bi \)
The imaginary unit forms the basis for complex number calculations, helping us perform operations that were previously impossible with just real numbers.

Simplifying Radicals

Simplifying radicals, especially when they involve negative numbers, requires careful attention to how the imaginary unit is applied. When dealing with radicals, always start by inspecting whether there's a negative present under the square root.
- For \( \sqrt{-a} \), you would start with recognizing it as \( \sqrt{a}i \)- In the original problem, \( \sqrt{-7} = \sqrt{7}i \) and \( \sqrt{-49} = 7i \) due to the negative sign
Simplification also includes breaking down composite radicals into their simplest form. For example, \( \sqrt{28} \) simplifies to \( 2\sqrt{7} \) because it can be factored into \( 4 \times 7 \), and \( \sqrt{4} = 2 \).
By simplifying these expressions, calculations become much more manageable.

Division of Complex Numbers

Dividing complex numbers might seem tricky, but it's an extension of basic division. When dividing expressions where complex numbers are involved, it’s important to simplify both the numerator and the denominator first.
In this exercise, when we reached the division step, it was:

• Numerator: \(-7\sqrt{7}\)
• Denominator: \(2\sqrt{7}\)

As you can see, the \(\sqrt{7}\) in both expressions canceled each other.
- After cancelation, you only need to deal with the coefficients, resulting in \(-\frac{7}{2}\)
Dividing complex numbers can also involve multiplying by the conjugate if necessary, but in this case, the direct simplification was enough since we were not dealing with a complex number in standard form.

Purely Real Number

A purely real number is part of the complex number system but with no imaginary portion. It is written in the form \( a + 0i \). This signifies that the coefficient of the imaginary part \( (b) \) is zero.
From the problem's solution, the final result \(-\frac{7}{2} + 0i \) is purely real:

• The imaginary part: \(0i\), shows there's no contribution from \(i\) needed here.
• The real part: \(-\frac{7}{2}\), stands alone without any imaginary component.

Purely real numbers often emerge in problems involving complex numbers, especially when imaginary units cancel out during calculations. These numbers play a crucial role in understanding the full scope of how complex numbers work and are fundamental when comparing to exclusively real numbers.