Problem 43 - Precalculus Mathematics for Calculus

Question

Evaluate the expression and write the result in the form \(a+b i\) $$\frac{4+6 i}{3 i}$$

Step-by-Step Solution

Answer

The expression simplifies to \(2 - \frac{4}{3}i\).

Step 1: Multiply by the Conjugate of the Denominator

To simplify the expression \(\frac{4 + 6i}{3i}\), we multiply both numerator and denominator by the conjugate of the denominator. Since the denominator is \(3i\), its conjugate is \(-3i\). This will help us eliminate the imaginary part in the denominator.Multiply:\[\frac{(4 + 6i) \cdot (-3i)}{3i \cdot (-3i)}\]

Step 2: Simplify the Denominator

Next, we need to simplify \(3i \cdot (-3i)\). We know that \(i^2 = -1\), so:\[3i \cdot (-3i) = -9i^2 = 9\]Thus, the denominator becomes 9.

Step 3: Simplify the Numerator

Now, simplify the numerator:\((4 + 6i) \cdot (-3i) = 4 \cdot (-3i) + 6i \cdot (-3i)\).Calculate each part:- \(4 \cdot (-3i) = -12i\)- \(6i \cdot (-3i) = -18i^2\)Since \(i^2 = -1\), we have:\(-18i^2 = 18\).So the numerator simplifies to \(18 - 12i\).

Step 4: Divide Each Term by the Denominator

With simplified numerator and denominator, divide each term in the numerator by the denominator:\[\frac{18 - 12i}{9} = \frac{18}{9} - \frac{12i}{9}\]This gives us:\[2 - \frac{12i}{9} = 2 - \frac{4i}{3}\].

Step 5: Write the Expression in the Form \(a + bi\)

The expression is now in the form \(a + bi\), where \(a = 2\) and \(b = -\frac{4}{3}\). Thus, the simplified expression is:\[2 - \frac{4}{3}i\].

Key Concepts

Imaginary Unit

Complex numbers have a real part and an imaginary part, usually represented as \(a + bi\), where \(i\) is the imaginary unit. The imaginary unit is defined as \(i^2 = -1\). This special property of \(i\) is what distinguishes complex numbers from real numbers. In any multiplication involving \(i\), such as \(6i \cdot (-3i)\), you apply this rule: since \(i^2 = -1\), then \(6i \cdot (-3i) = -18i^2 = 18\). Understanding \(i\) and its properties is fundamental to working with complex numbers.

Complex Conjugate

The complex conjugate of a complex number \(a + bi\) is \(a - bi\). When you multiply a complex number by its conjugate, you get a real number. This is useful in simplifying expressions, especially when you need to "rationalize" denominators containing imaginary terms. For example, the conjugate of \(3i\) is \(-3i\), and multiplying \(3i\) by \(-3i\) gives you a real result: \(-9i^2 = 9\). This step of multiplying by the conjugate is essential to eliminating the imaginary part from the denominator, making complex fractions easier to work with.

Rationalizing the Denominator

Rationalizing the denominator involves rewriting a fraction so that there are no imaginary numbers in the denominator. This process often involves multiplying both the numerator and denominator by the conjugate of the denominator.

• This creates a real number in the denominator like when multiplying \(3i\) by \(-3i\) results in 9.
• It simplifies the fraction into a form that's easier to read and work with, such as turning \(\frac{4+6i}{3i}\) into \(2 - \frac{4}{3}i\).

Simplifying in this way does not change the value of the fraction, but it aligns the expression with mathematical convention.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves performing operations systematically to reduce expressions to their simplest form. This includes distributing, combining like terms, and reducing fractions. In our example, to simplify \((4 + 6i) \cdot (-3i)\), you distribute \(-3i\) to each term in the parenthesis, yielding \(-12i - 18i^2\). Since \(i^2 = -1\), \(-18i^2\) becomes 18, and combined with \(-12i\), the expression simplifies to \(18 - 12i\). Once you have a simplified expression, you divide each term by the denominator to complete the simplification: \(\frac{18 - 12i}{9}\) results in \(2 - \frac{4}{3}i\). Understanding each step ensures clarity and accuracy when working with complex numbers.