Questions: Write the expression in the standard form a + bi. (8-i)/(7+i) (8-i)/(7+i) = (Simplify your answer. Type your answer in the form a + bi.)

Transcript text: Write the expression in the standard form a $+b i$. \[ \frac{8-i}{7+i} \] $\frac{8-i}{7+i}=$ $\square$ (Simplify your answer. Type your answer in the form a + bi.)

Solution

Solution Steps

Step 1: Multiply the numerator and the denominator by the conjugate of the denominator

To divide the complex numbers $8 - i$ and $7 + i$, we multiply both by the conjugate of the denominator $7 - i$. This results in: \[\frac{(8 - i)(7 - i)}{7^2 + 1^2}\]

Step 2: Apply the distributive property (foil method)

The numerator becomes $8 \cdot 7 - 1 \cdot i + 1 \cdot 7i - 8 \cdot i^2$. Since $i^2 = -1$, this simplifies to: $55 - 15i$.

Step 3: Simplify the expression

The denominator simplifies to $50$. So, the division gives us $1.1 - 0.3i$.