To simplify the expression [tex]\(\frac{2 + 7i}{3 - 3i}\)[/tex] and express it in the standard form [tex]\(a + bi\)[/tex], we will use the method of multiplying the numerator and the denominator by the conjugate of the denominator. Here are the detailed steps:
1. Identify the Numerator and the Denominator:
The numerator is [tex]\(2 + 7i\)[/tex] and the denominator is [tex]\(3 - 3i\)[/tex].
2. Find the Conjugate of the Denominator:
The conjugate of [tex]\(3 - 3i\)[/tex] is [tex]\(3 + 3i\)[/tex].
3. Multiply the Numerator and the Denominator by the Conjugate of the Denominator:
[tex]\[
\frac{2 + 7i}{3 - 3i} \times \frac{3 + 3i}{3 + 3i} = \frac{(2 + 7i)(3 + 3i)}{(3 - 3i)(3 + 3i)}
\][/tex]
4. Simplify the Denominator:
The denominator can be simplified using the difference of squares:
[tex]\[
(3 - 3i)(3 + 3i) = 3^2 - (3i)^2 = 9 - 9i^2
\][/tex]
Since [tex]\(i^2 = -1\)[/tex], this becomes:
[tex]\[
9 - 9(-1) = 9 + 9 = 18
\][/tex]
5. Expand the Numerator:
Use the distributive property to expand [tex]\((2 + 7i)(3 + 3i)\)[/tex]:
[tex]\[
(2 + 7i)(3 + 3i) = 2 \cdot 3 + 2 \cdot 3i + 7i \cdot 3 + 7i \cdot 3i
\][/tex]
This simplifies to:
[tex]\[
6 + 6i + 21i + 21i^2
\][/tex]
Since [tex]\(i^2 = -1\)[/tex], we get:
[tex]\[
6 + 6i + 21i + 21(-1) = 6 + 27i - 21 = -15 + 27i
\][/tex]
6. Combine Numerator and Denominator:
Now we have:
[tex]\[
\frac{-15 + 27i}{18}
\][/tex]
7. Separate the Real and Imaginary Parts:
Divide both the real part and the imaginary part by 18:
[tex]\[
\frac{-15}{18} + \frac{27i}{18}
\][/tex]
Simplify the fractions:
[tex]\[
-\frac{5}{6} + \frac{3}{2}i
\][/tex]
In conclusion, when we simplify [tex]\(\frac{2 + 7i}{3 - 3i}\)[/tex], we get the complex number in its simplest form:
[tex]\[
-\frac{5}{6} + \frac{3}{2}i
\][/tex]
Using numerical evaluation, we can identify that this corresponds to approximately:
[tex]\[
-0.8333333333333334 + 1.5i
\][/tex]
So, the simplest form of the expression [tex]\(\frac{2 + 7i}{3 - 3i}\)[/tex] is [tex]\(a + bi = -\frac{5}{6} + \frac{3}{2}i\)[/tex].
