Problem 34 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-34: Write the expression in the form \(a+b i\), where \(a\) and \(b\) are real numbers. $$ \frac{\sqrt{-25}}{\sqrt{-16} \sqrt{-81}} $$

Step-by-Step Solution

Answer

The expression simplifies to \(0 - \frac{5}{36}i\).

Step 1: Simplifying the Numerator and Denominators

To simplify the expression, we start by simplifying each radical expression. The square root of a negative number can be rewritten using imaginary numbers. The square root of \(-25\) is \(\sqrt{-25} = \sqrt{25} \, \cdot \, \sqrt{-1} = 5i\). Similarly, we simplify the denominators: \(\sqrt{-16} = 4i\) and \(\sqrt{-81} = 9i\).

Step 2: Simplifying the Entire Expression

With these simplifications, the original expression becomes \(\frac{5i}{(4i) \cdot (9i)}\). Simplifying the denominator, \((4i) \cdot (9i) = 36i^2 = 36(-1) = -36\). So, the expression is now \(\frac{5i}{-36}\).

Step 3: Rewriting in the Form \(a+bi\)

Now, divide the imaginary part: \(\frac{5i}{-36} = -\frac{5}{36}i\). In the form \(a + bi\), the real part \(a = 0\) and the imaginary part \(b = -\frac{5}{36}\). Thus, the expression simplifies to \(0 - \frac{5}{36}i\).

Key Concepts

Imaginary Numbers

Imaginary numbers might sound mysterious, but they are quite approachable once you understand their purpose. An imaginary number is a type of complex number that includes the unit imaginary number, denoted as \(i\), which is defined as \(i = \sqrt{-1}\). This allows us to take square roots of negative numbers — a task that is impossible within the realm of real numbers. For instance, when you encounter \(\sqrt{-25}\), you can express it using \(i\) by recognizing that it's the square root of a positive number multiplied by \(i\). Specifically, \(\sqrt{-25} = \sqrt{25} \, \cdot \, \sqrt{-1} = 5i\). This decomposition helps in handling negative roots and is essential in the study of complex numbers.**Why Imaginary Numbers?**

• Enable calculations involving square roots of negative numbers.
• Expand calculations and solutions to equations that lack real solutions.
• Use in advanced fields such as engineering and physics.

Imaginary numbers extend the number system, allowing more equations to have solutions.

Radical Expressions

Radical expressions involve roots, such as the square root, cube root, etc. Simplifying radical expressions is key to making calculations easier and solving algebraic equations. When dealing with negative numbers under a square root, as seen in expressions like \(\sqrt{-16}\), it becomes necessary to use imaginary numbers for simplification. To simplify such expressions:

• Identify and separate the negative sign by using \(i\). For example, \(\sqrt{-16} = \sqrt{16} \, \cdot \, \sqrt{-1} = 4i\).
• Apply any arithmetic operations, recognizing that \(i^2=-1\).
• Combine terms when possible to reduce complexity.

Radical expressions form a crucial part of algebra, allowing you to express roots more clearly, especially when complex numbers are involved.

Simplifying Expressions

Simplifying expressions involves reducing the expression to its most basic form while preserving equivalence. This process often involves combining like terms, performing arithmetic operations, and reducing fractions wherever feasible. In expressions involving imaginary numbers, like the one given in the exercise, the goal is often to represent the expression in the standard form of complex numbers \(a + bi\).The expression \(\frac{\sqrt{-25}}{\sqrt{-16} \cdot \sqrt{-81}}\) was simplified step-by-step in the solution:

• Each radical component was expressed with \(i\), resulting in \(\frac{5i}{4i \, \cdot \, 9i}\).
• Multiplication of the denominator yielded \(36i^2\), replaced by \(-36\) due to \(i^2 = -1\).
• Finally, by dividing \(\frac{5i}{-36}\), it’s rewritten in the form \(a + bi\) as \(0 - \frac{5}{36}i\).

This approach ensures expressions are not only simplified but also ready for further analysis or application in different mathematical fields.