Problem 47 - College Algebra and Calculus: An Applied Approach

Question

Write the quotient in standard form. $$\frac{5}{4-2 i}$$

Step-by-Step Solution

Answer

The quotient in standard form is \(\frac{5}{3} + \frac{5i}{6}\)

Step 1: Identification of Conjugate

Firstly, identify the conjugate of the denominator. The conjugate of a complex number \(a + bi\) is \(a - bi\). So, the conjugate of \(4 - 2i\) is \(4 + 2i\).

Step 2: Multiplication with Conjugate

Next, multiply the numerator and the denominator by this conjugate such that: \[ \frac{5}{4-2 i} \times \frac{4+2i}{4+2i} \]

Step 3: Distributive Property

Now apply the distributive property separately in the numerator and the denominator. The numerator would be \(20 + 10i\) and the denominator would be \(16 + 8i - 8i -4\) after simplifying.

Step 4: Simplification

Simplify the expressions to get the (a+bi) form. The numerator remains the same, but the denominator simplifies to \(16 - 4 = 12\), so the final expression becomes \( \frac{20 + 10i}{12}\).

Step 5: Divide out common factor

Finally, you divide out the common factor to get the expression in lowest terms. The common factor is 2. So, you divide each part of the numerator and the denominator by 2: \( \frac{10 + 5i}{6}\) or \(\frac{5}{3} + \frac{5i}{6}\).