Problem 55 - Algebra for College Students

Question

Write each of the following in terms of \(i\), perform the indicated operations, and simplify. For example, $$ \begin{aligned} \sqrt{-3} \sqrt{-8} &=(i \sqrt{3})(i \sqrt{8}) \\ &=i^{2} \sqrt{24} \\ &=(-1) \sqrt{4} \sqrt{6} \\ &=-2 \sqrt{6} \end{aligned} $$ $$\frac{\sqrt{-25}}{\sqrt{-4}}$$

Step-by-Step Solution

Answer

The simplified form is \(2.5\).

Step 1: Express Each Term with i

To begin simplifying, express each of the square root terms using the imaginary unit \(i\). The expression becomes: \( \frac{\sqrt{-25}}{\sqrt{-4}} = \frac{i \sqrt{25}}{i \sqrt{4}} \).

Step 2: Simplify the Square Roots

Next, calculate the square roots of the positive numbers in the numerator and denominator: \( \sqrt{25} = 5 \) and \( \sqrt{4} = 2 \). Substitute these values back into the expression, yielding \( \frac{i \cdot 5}{i \cdot 2} \).

Step 3: Simplify the Fraction

Since both the numerator and denominator have the imaginary unit \(i\), they can be cancelled out, simplifying the expression to \( \frac{5}{2} \).

Step 4: Final Simplification

The expression simplifies to just a real number: \( \frac{5}{2} = 2.5 \).

Key Concepts

Complex Numbers

Complex numbers are a combination of real and imaginary numbers. They have the form \( a + bi \), where \( a \) is the real part, and \( bi \) is the imaginary part. The imaginary unit \( i \) is defined as \( \sqrt{-1} \), which is the basis for all imaginary numbers.
To grasp this concept better, think of complex numbers as points on a plane, with the real part corresponding to the horizontal axis and the imaginary part to the vertical axis.

• The complex plane, often referred to as the Argand plane, provides a way to visualize complex numbers.
• Every complex number can be uniquely represented as a point in this plane or as a vector from the origin.
• Operations such as addition and subtraction of complex numbers can be visualized as vector addition and subtraction.

Understanding complex numbers is crucial as they appear in various fields, such as engineering, physics, and even finance, mainly due to their ability to solve equations that do not have solutions within the real number system.

Simplifying Radicals

Simplifying radicals involves expressing a radical expression in its simplest form. Radicals are often simplified to eliminate any factor that can be squared and extracted from under the radical sign.
Here's a quick look at how to simplify radicals:

• Factor the number under the radical sign into its prime factors.
• Pair the same factors and move them outside the radical.
• If a number is repeated twice under the radical, it can be moved outside as a single number.
• Simplifying helps in performing operations like addition, subtraction, and multiplication more easily.

For example, the radical \( \sqrt{24} \) can be simplified by breaking it down to \( \sqrt{4 \times 6} \) which simplifies to \( 2\sqrt{6} \). Simplifying radicals is a key skill, especially when dealing with complex or large polynomials.

Square Roots

Square roots are the numbers that, when multiplied by themselves, give the original number. For example, the square root of 25 is 5 because \( 5 \times 5 = 25 \).
In the context of imaginary numbers, finding square roots involves using the imaginary unit \( i \). For negative numbers, the square root is often expressed in terms of \( i \):

• The square root of \(-1\) is defined as \( i \).
• For any negative number \( -n \), its square root can be expressed as \( i\sqrt{n} \).
• This is why \( \sqrt{-25} \) becomes \( i\sqrt{25} \).

Understanding square roots, especially with negative numbers, is crucial as it often serves as the basis for operations involving complex numbers.

Algebraic Operations

Algebraic operations with complex numbers follow similar principles as operations with real numbers but require special consideration of the imaginary unit \( i \).

• Addition and Subtraction: Combine the real parts and the imaginary parts separately. For example, \((3 + 4i) + (1 + 2i) = 4 + 6i\).
• Multiplication: Use the distributive property and remember that \( i^2 = -1 \). For example, \((3 + 2i)(1 + 4i) = 3 + 12i + 2i + 8i^2 = 3 + 14i - 8 = -5 + 14i\).
• Division: Simplify by multiplying the numerator and the denominator by the complex conjugate of the denominator.

In exercises involving imaginary numbers, such as the given problem, algebraic operations will often involve simplifying by using properties of \( i \), such as \( i^2 = -1 \), which can help simplify expressions to real numbers, as seen in the provided steps.