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Chapter 7: Problem 406

Find the \(5^{\text {th }}\) term of the expansion of \((3 \mathrm{y}-4 \mathrm{w})^{8}\).

Short Answer

Expert verified
The 5th term of the expansion of \((3y - 4w)^8\) is \(143360y^4w^4\).

Step by step solution

01

Calculate the binomial coefficient

Calculate \({n \choose r} = {8 \choose 4}\) using the formula \(\frac{n!}{r!(n-r)!}\). \(\binom{8}{4} = \frac{8!}{4!(8-4)!} = \frac{8!}{4!4!} = \frac{8\times7\times6\times5}{4\times3\times2\times1} = 70\).
02

Plug the values into the binomial theorem formula

Now that we have the binomial coefficient, we can plug in the values into the binomial theorem formula \({n \choose r}\;a^{n-r}b^{r}\) for our expansion. In our case, \({n \choose r}=70\), \(a=3y\), \(b=-4w\), \(n-r=4\), and \(r=4\). So, the 5th term is \(T_5=70\;(3y)^{4}(-4w)^{4}\).
03

Simplify the expression

Now, it's a matter of simplifying the expression. \(T_5 = 70\;(3^4y^4)(-4^4w^4)\) \(T_5 = 70\;(81y^4)(256w^4)\) \(T_5 = 70\;\times\;81\;\times\;256\;y^4w^4\) Multiplying the coefficients gives \(T_5 = 143360\;y^4w^4\) The 5th term of the expansion of \((3y - 4w)^8\) is \(143360y^4w^4\).

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