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Chapter 10.5

Q. 10.5.4

Finding a Single Term of a Binomial Expansion

Find the fourth term in the expansion of (3 x+2 y)^7.

Step-by-Step

Verified Solution

We are looking for the fourth term. The value of r is one less than the term to be found. Thus, r = 3. In the expansion of (3 x+2 y)^7, a = 3x, b = 2y, and n = 7.

The fourth term is

Now we need to evaluate the factorial expression and raise 3x and 2y to the indicated powers. We obtain

\frac{7 !}{3 ! 4 !}\left(81 x^4\right)\left(8 y^3\right)=\frac{7 \cdot 6 \cdot 5 \cdot \cancel{4 !}}{3 \cdot 2 \cdot 1 \cdot \cancel{4 !}}\left(81 x^4\right)\left(8 y^3\right)=35\left(81 x^4\right)\left(8 y^3\right)=22,680 x^4 y^3.

The fourth term of (3 x+2 y)^7 is 22,680 x^4 y^3.