Questions: Simplify and write your answer in the form (a+bi). [ frac52+i ] (10+i) (2+i) (frac10-5i5) (2-i)

Solution Steps

To simplify the expression \(\frac{5}{2+i}\) and write it in the form \(a + bi\), we need to multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of \(2 + i\) is \(2 - i\). This will help us eliminate the imaginary part in the denominator.

We need to simplify the expression \(\frac{5}{2+i}\) and write the answer in the form \(a + bi\).

To simplify \(\frac{5}{2+i}\), we need to rationalize the denominator. We do this by multiplying the numerator and the denominator by the conjugate of the denominator. The conjugate of \(2+i\) is \(2-i\).

\[ \frac{5}{2+i} \cdot \frac{2-i}{2-i} = \frac{5(2-i)}{(2+i)(2-i)} \]

Next, we simplify the denominator \((2+i)(2-i)\):

\[ (2+i)(2-i) = 2^2 - i^2 = 4 - (-1) = 4 + 1 = 5 \]

Now, we simplify the numerator \(5(2-i)\):

\[ 5(2-i) = 10 - 5i \]

Putting it all together, we have:

\[ \frac{5(2-i)}{5} = \frac{10 - 5i}{5} = 2 - i \]

Final Answer