Of course! Let's go through the detailed, step-by-step solution to expand the expression [tex]\(\left(\frac{2}{3} x y - 3\right)^2\)[/tex].
1. Identify the expression:
The expression to expand is [tex]\(\left(\frac{2}{3} x y - 3\right)^2\)[/tex].
2. Apply the binomial theorem:
The binomial theorem states that [tex]\((a - b)^2 = a^2 - 2ab + b^2\)[/tex]. Here, [tex]\(a = \frac{2}{3} x y\)[/tex] and [tex]\(b = 3\)[/tex].
3. Calculate [tex]\(a^2\)[/tex]:
[tex]\[
a^2 = \left(\frac{2}{3} x y\right)^2 = \left(\frac{2}{3}\right)^2 x^2 y^2 = \frac{4}{9} x^2 y^2
\][/tex]
4. Calculate [tex]\(-2ab\)[/tex]:
[tex]\[
-2ab = -2 \cdot \left(\frac{2}{3} x y\right) \cdot 3 = -2 \cdot \frac{6}{3} x y = -4 x y
\][/tex]
5. Calculate [tex]\(b^2\)[/tex]:
[tex]\[
b^2 = 3^2 = 9
\][/tex]
6. Combine the results:
[tex]\[
\left(\frac{2}{3} x y - 3\right)^2 = \frac{4}{9} x^2 y^2 - 4 x y + 9
\][/tex]
7. Interpret the result from above steps:
We can see that the expanded form of the given expression is:
[tex]\[
0.444444444444444 x^2 y^2 - 4.0 x y + 9
\][/tex]
Therefore, the expanded expression [tex]\(\left(\frac{2}{3} x y - 3\right)^2\)[/tex] is [tex]\(0.444444444444444 x^2 y^2 - 4.0 x y + 9\)[/tex].
