Problem 25 - Algebra and Trigonometry

Question

Divide and express the result in standard form. $$ \frac{8 i}{4-3 i} $$

Step-by-Step Solution

Answer

The result of the division in standard form is \(-\frac{24}{25} + \frac{32}{25}i\).

Step 1: Identify and Multiply by the Conjugate

The conjugate of \(4 - 3i\) is \(4 + 3i\). Multiply the numerator and the denominator of the fraction by this conjugate:\[ \frac{8i * (4 + 3i)}{(4 - 3i) * (4 + 3i)}\]

Step 2: Simplify the Numerator and Denominator

Simplify the multiplication you performed in the last step. You can distribute the factor in the numerator. For the denominator difference of squares formula can be used which states that for any real numbers a and b \((a - b)(a + b) = a^2 - b^2\):Numerator simplification: \[8i * (4 + 3i) = 32i + 24i^2\]Remember that \(i^2 = -1\), so you can replace \(24i^2\) with \(-24\). Hence the numerator simplifies to \(32i -24\).Denominator simplification using the formula (\(a^2 - b^2\)):\[(4 - 3i) * (4 + 3i) = 4^2 - (3i)^2 = 16 - (9 * -1) = 16 + 9 = 25\]Hence, the fraction simplifies to:\[\frac{32i -24}{25}\]

Step 3: Divide the Numerator by the Denominator

It's best to write complex numbers in the form a + bi. Divide both the real and imaginary parts of the numerator by the denominator to achieve this:\[-\frac{24}{25} + \frac{32}{25}i\]