Problem 65 - Algebra and Trigonometry Real Mathematics, Real People

Question

Perform the operation and write the result in standard form. $$\frac{i}{3-2 i}+\frac{2 i}{3+8 i}$$

Step-by-Step Solution

Answer

The result in standard form of the given operation \(\frac{i}{3-2 i}+\frac{2 i}{3+8i}\) is \(0.15 - 0.14i\)

Step 1: Rationalize the denominators

In order to rationalize the denominator, multiply both the numerator and denominator by the conjugate of the denominator of the fractions. \nThe conjugate of \(3 - 2i\) is \(3 + 2i\) and of \(3 + 8i\) is \(3 - 8i\). Hence, \(\frac{i}{3-2 i} \times \frac{3+2i}{3+2i} + \frac{2 i}{3+8 i} \times \frac{3-8i}{3-8i}\)

Step 2: Simplify the expressions

Now, simplify the expressions obtained in step 1 using the rule \(i^2=-1\) and the property \((a + bi)(a - bi)=a^2 + b^2\).\nAfter simplification, you will get \(\frac{3i+2}{13} + \frac{6-16i}{73}\)

Step 3: Combine the fractions

Next, write these complex numbers in standard form: \(\frac{2}{13} + \frac{3i}{13} +\frac{6}{73} -\frac{16i}{73}\)

Step 4: Combine Like Terms

After simplifying, combine like terms.\nThe answer will be: \(0.15 - 0.14i\) which is the standard form of the result.