Question6. What is the coefficient of $1 / x^{6}$ in the expression $\left(x^{5}+1 / x^{3}\right)^{10}$ .
ANS:

Studdy Solution

STEP 1

1. We are dealing with a binomial expansion of the form $(a + b)^n$ .
2. The binomial theorem can be used to expand $(x^5 + \frac{1}{x^3})^{10}$ .
3. We need to find the specific term in the expansion that results in $x^{-6}$ .

STEP 2

1. Identify the general term in the binomial expansion.
2. Determine the power of $x$ in each term.
3. Set the power of $x$ equal to $-6$ .
4. Solve for the binomial coefficient.
5. Calculate the coefficient of the desired term.

STEP 3

Identify the general term in the binomial expansion of $(x^5 + \frac{1}{x^3})^{10}$ .
The general term in the expansion of $(a + b)^n$ is given by:
$T_k = \binom{n}{k} a^{n-k} b^k$
For our expression, $a = x^5$ , $b = \frac{1}{x^3}$ , and $n = 10$ .
Thus, the general term $T_k$ is:
$T_k = \binom{10}{k} (x^5)^{10-k} \left(\frac{1}{x^3}\right)^k$

STEP 4

Determine the power of $x$ in each term.
$T_k = \binom{10}{k} x^{5(10-k)} \cdot x^{-3k}$
Combine the exponents of $x$ :
$T_k = \binom{10}{k} x^{50 - 5k - 3k} = \binom{10}{k} x^{50 - 8k}$

STEP 5

Set the power of $x$ equal to $-6$ to find the appropriate term.
$50 - 8k = -6$

STEP 6

Solve for $k$ .
$50 - 8k = -6$ $50 + 6 = 8k$ $56 = 8k$ $k = \frac{56}{8} = 7$

STEP 7

Calculate the coefficient of the desired term.
The coefficient is given by $\binom{10}{7}$ .
Calculate $\binom{10}{7}$ :
$\binom{10}{7} = \binom{10}{3} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120$
The coefficient of $\frac{1}{x^6}$ in the expression $\left(x^{5}+1 / x^{3}\right)^{10}$ is $\boxed{120}$ .

Was this helpful?

Get Studdy

Scan QR code with your device to get Studdy on iOS or Android

QR code

Studdy solves anything!

Download the app

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Parents Influencer program Contact Policy Terms