Question

Solve the complex number (35i+10)/(40i+64)

Answer 1

$\frac{255}{712}+\frac{230}{712}i$

Answer 2

Solution:

We wish to simplify

$$\frac{35i + 10}{40i + 64} = \frac{10 + 35i}{64 + 40i}.$$

Multiply numerator and denominator by the conjugate of the denominator, $64-40i$:

$$\frac{(10+35i)(64-40i)}{(64+40i)(64-40i)}.$$

Step 1: Numerator Calculation

$$(10+35i)(64-40i)= 10\cdot64 + 10\cdot(-40i) + 35i\cdot64 + 35i\cdot(-40i)$$ $$= 640 - 400i + 2240i - 1400i^2.$$

Since $i^2=-1$, we have:

$$-1400i^2 = 1400.$$

Thus, the numerator becomes:

$$640 + 1400 + (2240i-400i)= 2040 + 1840i.$$

Step 2: Denominator Calculation

$$(64+40i)(64-40i)= 64^2 - (40i)^2=4096 - 1600i^2.$$

Using $i^2=-1$:

$$4096 - 1600(-1)=4096+1600=5696.$$

Step 3: Write in $a+bi$ Form and Simplify

Thus, we have:

$$\frac{2040+1840i}{5696}.$$

Divide numerator and denominator by 8:

$$\frac{2040\div 8 + 1840i\div 8}{5696\div 8} = \frac{255+230i}{712}.$$

So, the expression in $a+bi$ form is:

$$\frac{255}{712}+\frac{230}{712}i.$$