Express the following in the form a+ bi
`(5)/(2i-7i^(2))`

Answer

To express the complex number \( \frac{5}{2i - 7i^2} \) in the form \( a + bi \), we will follow these steps:

### Step 1: Simplify the denominator
First, we need to simplify the denominator \( 2i - 7i^2 \). We know that \( i^2 = -1 \), so we can substitute this into the expression:

\[ 7i^2 = 7(-1) = -7 \]

Thus, the denominator becomes:

\[ 2i - 7i^2 = 2i + 7 \]

### Step 2: Rewrite the expression
Now we can rewrite the expression:

\[ z = \frac{5}{2i + 7} \]

### Step 3: Multiply by the conjugate
To eliminate the imaginary part from the denominator, we multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of \( 2i + 7 \) is \( 7 - 2i \):

\[ z = \frac{5(7 - 2i)}{(2i + 7)(7 - 2i)} \]

### Step 4: Expand the numerator and denominator
Now we expand both the numerator and the denominator:

**Numerator:**
\[ 5(7 - 2i) = 35 - 10i \]

**Denominator:**
Using the formula \( (a + b)(a - b) = a^2 - b^2 \):

\[ (2i + 7)(7 - 2i) = 7^2 - (2i)^2 = 49 - 4i^2 \]
Substituting \( i^2 = -1 \):

\[ 49 - 4(-1) = 49 + 4 = 53 \]

### Step 5: Combine the results
Now we can combine the results:

\[ z = \frac{35 - 10i}{53} \]

### Step 6: Separate into real and imaginary parts
We can separate this into real and imaginary parts:

\[ z = \frac{35}{53} - \frac{10}{53}i \]

### Final Result
Thus, we have expressed the complex number in the form \( a + bi \):

\[ z = \frac{35}{53} - \frac{10}{53}i \]