Express the following in the form a+ bi
`(2+i)/((3-i)(1+2i))`

Answer

To express the given complex fraction \((2+i)/((3-i)(1+2i))\) in the form \(a + bi\), we will follow these steps:

### Step 1: Simplify the Denominator
First, we need to simplify the denominator \((3 - i)(1 + 2i)\).

\[ (3 - i)(1 + 2i) = 3 \cdot 1 + 3 \cdot 2i - i \cdot 1 - i \cdot 2i \]
\[ = 3 + 6i - i - 2i^2 \]
Since \(i^2 = -1\), we can replace \( -2i^2 \) with \( +2 \):
\[ = 3 + 6i - i + 2 = 5 + 5i \]

### Step 2: Rewrite the Expression
Now we can rewrite the original expression:
\[ \frac{2+i}{(3-i)(1+2i)} = \frac{2+i}{5+5i} \]

### Step 3: Rationalize the Denominator
To express this in the form \(a + bi\), we need to rationalize the denominator. We do this by multiplying the numerator and denominator by the conjugate of the denominator:
\[ \frac{2+i}{5+5i} \cdot \frac{5-5i}{5-5i} = \frac{(2+i)(5-5i)}{(5+5i)(5-5i)} \]

### Step 4: Calculate the Denominator
Now, calculate the denominator:
\[ (5+5i)(5-5i) = 5^2 - (5i)^2 = 25 - 25(-1) = 25 + 25 = 50 \]

### Step 5: Calculate the Numerator
Now calculate the numerator:
\[ (2+i)(5-5i) = 2 \cdot 5 + 2 \cdot (-5i) + i \cdot 5 + i \cdot (-5i) \]
\[ = 10 - 10i + 5i - 5i^2 \]
Again, replace \(-5i^2\) with \(+5\):
\[ = 10 - 10i + 5 + 5 = 15 - 5i \]

### Step 6: Combine Results
Now we can combine the results:
\[ \frac{15 - 5i}{50} = \frac{15}{50} - \frac{5i}{50} = \frac{3}{10} - \frac{1}{10}i \]

### Final Answer
Thus, the expression in the form \(a + bi\) is:
\[ \frac{3}{10} - \frac{1}{10}i \]

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