Problem 80 - Elementary and Intermediate Algebra

Question

Simplify. Write the result in the form \(a+b i .\) \(\frac{6+7 i}{3-4 i}\)

Step-by-Step Solution

Step 1: Apply the relevant algebraic rules

Use properties of exponents, radicals, or algebraic identities to simplify the expression.

Step 2: State the simplified result

The simplified expression is obtained by combining like terms and reducing.

Key Concepts

Simplifying Expressions

When we simplify expressions, our goal is to make them easier to read and understand. In the context of complex numbers, simplifying often means rewriting expressions so that they don't include fractions with complex denominators. Complex numbers come in the form of \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. The imaginary unit is represented by \(i\), where \(i^2 = -1\).

To simplify the expression \( \frac{6+7i}{3-4i} \), we'll need to eliminate the imaginary component from the denominator. This simplifies our expression, making calculations easier and more straightforward. A key technique used for this process is multiplying both the numerator and the denominator by the complex conjugate of the denominator.

Imaginary Numbers

Imaginary numbers are a fascinating component of complex numbers. These numbers are defined based on the imaginary unit \(i\), where \(i\) is the square root of \(-1\). Though they are called "imaginary," these numbers are very real in mathematics and physics, helping to solve equations that don't have solutions among real numbers.

When you see an expression like \(6 + 7i\), \(7i\) is the imaginary part. Imaginary numbers extend our number system, enabling us to tackle a wider variety of mathematical problems. Though they can't be plotted on a typical number line, they can be represented on a complex plane, where the x-axis denotes real numbers and the y-axis denotes imaginary numbers. This combined representation helps us to visualize and perform arithmetic on complex numbers efficiently.

Complex Conjugates

A complex conjugate is a powerful tool when dealing with complex numbers, particularly in simplifying expressions. The complex conjugate of a complex number flips the sign of the imaginary part, turning \(a + bi\) into \(a - bi\). It's particularly useful for removing imaginary components from denominators.

For example, the complex conjugate of \(3 - 4i\) is \(3 + 4i\). By multiplying both the numerator and denominator of a fraction by the complex conjugate of the denominator, you can "rationalize" the denominator. This means transforming it into a real number.

In our exercise, to simplify \( \frac{6+7i}{3-4i} \), we multiply by \( \frac{3+4i}{3+4i} \). This process not only simplifies the expression but also highlights how integral complex conjugates are to working with complex numbers efficiently.