Problem 28 - Algebra and Trigonometry

Question

Divide and express the result in standard form. $$ \frac{3-4 i}{4+3 i} $$

Step-by-Step Solution

Answer

The result is \(\frac{24}{25} - i\).

Step 1: Identify the Conjugate

Firstly, determine the conjugate of the denominator, which is \(4 - 3i\). The conjugate of a complex number flips the sign between the real and imaginary parts.

Step 2: Multiply by Conjugate

Multiply the numerator and denominator of the fraction by the conjugate of the denominator, \(4 - 3i\). This yields: \[ \frac{(3 - 4i) (4 - 3i)} {(4 + 3i) (4 - 3i)}\]

Step 3: Expand and Simplify

Expand and simplify the numerator and denominator separately. The numerator becomes \(3*4 + 3*(-3i) - 4*4i + 4i*(-3i)\), which simplifies to \(12 - 9i - 16i - 12i^2\). Because \(i^2 = -1\), substitute \(i^2\) with \(-1\), to get \(12 - 9i - 16i + 12 = 24 - 25i\). The denominator becomes \(4*4 + 4*(-3i) + 3i*4 - 3i*(-3i) = 16 -12i + 12i - 9i^2 = 16 + 9 = 25\), as the terms with \(i\) cancel out. Then the fraction becomes \( \frac{24 - 25i}{25}\).

Step 4: Express in standard form

The last step is to write the result in standard form, which is a + bi, where a and b are real numbers. This can be achieved by dividing both terms in the numerator by the denominator, receiving \(\frac{24}{25} - \frac{25i}{25}\). This reduces to \(\frac{24}{25} - i\).