To find the fifth term in the expansion of [tex]\((3x - 3y)^7\)[/tex], we can use the binomial theorem. The binomial theorem states that:
[tex]\[
(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k
\][/tex]
Here, [tex]\( a = 3x \)[/tex], [tex]\( b = -3y \)[/tex], and [tex]\( n = 7 \)[/tex].
The general term in this expansion is given by:
[tex]\[
T_{k+1} = \binom{n}{k} (3x)^{n-k} (-3y)^k
\][/tex]
We need to identify the fifth term in this expansion. Remember that the binomial coefficients are indexed starting from [tex]\( k = 0 \)[/tex]. Therefore, the fifth term corresponds to [tex]\( k = 4 \)[/tex].
Plugging [tex]\( k = 4 \)[/tex] into the general term formula, we get:
[tex]\[
T_5 = \binom{7}{4} (3x)^{7-4} (-3y)^4
\][/tex]
First, calculate the binomial coefficient [tex]\(\binom{7}{4}\)[/tex]:
[tex]\[
\binom{7}{4} = \frac{7!}{4!(7-4)!} = \frac{7!}{4!3!} = \frac{7 \cdot 6 \cdot 5 \cdot 4!}{4! \cdot 3 \cdot 2 \cdot 1} = 35
\][/tex]
Next, evaluate the powers of [tex]\( (3x) \)[/tex] and [tex]\( (-3y) \)[/tex]:
[tex]\[
(3x)^{7-4} = (3x)^3 = 27x^3
\][/tex]
[tex]\[
(-3y)^4 = (-3)^4 y^4 = 81y^4
\][/tex]
Now, combine these components to find [tex]\(T_5\)[/tex]:
[tex]\[
T_5 = 35 \cdot 27x^3 \cdot 81y^4
\][/tex]
[tex]\[
T_5 = 35 \cdot 2187 x^3 y^4 = 76545 x^3 y^4
\][/tex]
Thus, the fifth term in the expansion of [tex]\((3x - 3y)^7\)[/tex] is:
[tex]\[
76545 x^3 y^4
\][/tex]
