Ninth Term In (x-2y)^13: Binomial Expansion Explained

$$

Hey guys! Let's break down this binomial expansion problem step-by-step so you can totally nail it. We're looking for the ninth term in the expansion of $(x-2y)^{13}$. This might sound intimidating, but don’t worry, we’ll make it super clear. Understanding binomial expansions is super useful in math, and it's not as complicated as it looks once you get the hang of it. So, grab your pencils, and let's dive in!

Understanding Binomial Expansion

First off, what exactly is a binomial expansion? Simply put, it’s the process of expanding an expression in the form of $(a + b)^n$, where 'n' is a positive integer. When 'n' is small, you can multiply it out manually. But when 'n' gets bigger, like our case with 13, we need a more systematic approach. This is where the Binomial Theorem comes to the rescue. The Binomial Theorem gives us a formula to find any term in the expansion without having to expand the entire expression. Think of it as a shortcut, a magic trick to skip all the tedious work. And let’s be honest, who doesn’t love a good shortcut? We use combinations and powers to get there.

The Binomial Theorem Formula

The Binomial Theorem states that the (k+1)-th term in the expansion of $(a + b)^n$ is given by:

$T_{k+1} = \binom{n}{k} a^{n-k} b^k$

Where:

• $\binom{n}{k}$ is the binomial coefficient, also written as "n choose k", which equals $\frac{n!}{k!(n-k)!}$
• 'n' is the power to which the binomial is raised.
• 'k' is the term number minus 1 (because we start counting from 0).
• 'a' and 'b' are the terms inside the binomial.

This formula is your best friend for these kinds of problems. Memorize it, cherish it, and it will guide you through the binomial expansion wilderness. Now, let's break down each part of this formula so it feels less like a scary equation and more like a friendly set of instructions. We've got this!

Breaking Down the Formula

Let's dive deeper into each component of the formula. The binomial coefficient $\binom{n}{k}$ might look intimidating, but it’s just a way of counting the number of ways to choose 'k' items from a set of 'n' items. This concept is fundamental in combinatorics, which is all about counting different combinations and permutations. The exclamation marks in the formula represent factorials. For example, 5! (5 factorial) is 5 × 4 × 3 × 2 × 1 = 120. Factorials can get big really quickly, so using the binomial coefficient formula helps us simplify things before we get overwhelmed by massive numbers. Also, remember that 0! is defined as 1, which might seem weird, but it's super important for the formula to work correctly.

The terms 'a' and 'b' in the formula are simply the terms inside the binomial that you're expanding. In our case, 'a' is 'x' and 'b' is '-2y'. Pay close attention to the signs! A negative sign can make a big difference in your final answer. The exponents (n-k) and k tell you what power to raise each term to. Notice that the exponents always add up to 'n'. This is a handy way to double-check that you're doing things right. If they don't add up, something's gone sideways, and it's time to retrace your steps. This is a common mistake, so always keep an eye on those exponents. Understanding each piece of the formula makes the whole process way less daunting. You're not just plugging numbers into a black box; you're understanding why you're doing each step. And that's the key to mastering binomial expansions!

Applying the Formula to Our Problem

Now that we've got a solid grip on the theory, let’s apply this to our specific problem: finding the ninth term in the expansion of $(x - 2y)^{13}$. Remember, the goal is to use the Binomial Theorem formula to pluck out the exact term we need without having to slog through the entire expansion. This is where the magic happens, guys!

Identifying n, k, a, and b

First, we need to identify our values for n, k, a, and b from the given expression $(x - 2y)^{13}$.

• 'n' is the exponent, which is 13.
• Since we want the ninth term, k + 1 = 9, so k = 8. Remember, 'k' is always one less than the term number because we start counting from 0.
• 'a' is the first term inside the binomial, which is 'x'.
• 'b' is the second term inside the binomial, which is '-2y'. Don't forget the negative sign!

It's crucial to get these values right, or the whole calculation will go off the rails. Take your time to double-check each value before plugging them into the formula. A little bit of careful attention here can save you a lot of headaches later on.

Plugging the Values into the Formula

Now we plug these values into the Binomial Theorem formula:

$T_{9} = T_{8+1} = \binom{13}{8} x^{13-8} (-2y)^8$

Breaking it down:

• $\binom{13}{8}$ is the binomial coefficient we need to calculate.
• $x^{13-8}$ simplifies to $x^5$.
• $(-2y)^8$ is the term '-2y' raised to the power of 8.

This is where the calculation starts to get interesting. We're not just plugging in numbers anymore; we're starting to see how the formula transforms the original expression into the ninth term. Each part of this expression plays a crucial role in determining the final answer. Next, we'll crunch the numbers and simplify this expression to get to our solution.

Calculating the Binomial Coefficient and Simplifying

Okay, guys, let's get our hands dirty with some calculations! We need to compute the binomial coefficient $\binom{13}{8}$ and simplify the rest of the expression. This is where our arithmetic skills come into play, and a little bit of patience goes a long way. Don’t worry; we’ll take it step by step.

Calculating $\binom{13}{8}$

Remember the formula for the binomial coefficient:

$\binom{n}{k} = \frac{n!}{k!(n-k)!}$

So, for $\binom{13}{8}$, we have:

$\binom{13}{8} = \frac{13!}{8!(13-8)!} = \frac{13!}{8!5!}$

Now, let’s expand those factorials:

$\frac{13!}{8!5!} = \frac{13 \times 12 \times 11 \times 10 \times 9 \times 8!}{8! \times 5 \times 4 \times 3 \times 2 \times 1}$

Notice that we can cancel out the 8! in the numerator and denominator. This is a common trick that simplifies the calculation significantly. Always look for opportunities to cancel out factorials – it will save you a ton of time and effort.

After canceling, we get:

$\frac{13 \times 12 \times 11 \times 10 \times 9}{5 \times 4 \times 3 \times 2 \times 1}$

Now, we can simplify further by canceling out common factors:

• 5 × 2 = 10, so we can cancel out the 10 in the numerator and the 5 × 2 in the denominator.
• 4 × 3 = 12, so we can cancel out the 12 in the numerator and the 4 × 3 in the denominator.

This leaves us with:

$13 \times 11 \times 9 = 1287$

So, $\binom{13}{8} = 1287$. That's a big number, but we tackled it like pros! Calculating binomial coefficients can be a bit of a workout, but with practice, you'll get faster and more confident. The key is to break it down into smaller steps and look for opportunities to simplify.

Simplifying $x^{13-8} (-2y)^8$

Next, we need to simplify the rest of the expression:

$x^{13-8} (-2y)^8$

First, let's simplify the exponent:

$x^{13-8} = x^5$

Now, let's deal with $(-2y)^8$. Remember that when you raise a negative number to an even power, the result is positive:

$(-2y)^8 = (-2)^8 y^8 = 256y^8$

Putting it all together, we have:

$x^5 (256y^8) = 256x^5y^8$

We’ve successfully simplified this part of the expression. Notice how breaking it down into smaller pieces makes it much easier to manage. We handled the exponent, dealt with the negative sign, and raised the coefficient to the correct power. Each step is straightforward when you focus on one thing at a time.

Putting It All Together

Alright, team, we're in the home stretch! We've calculated the binomial coefficient and simplified the variable terms. Now, we just need to combine everything to find the ninth term in the expansion of $(x - 2y)^{13}$. This is the moment where all our hard work pays off!

Combining the Results

We found that:

• $\binom{13}{8} = 1287$
• $x^{13-8} (-2y)^8 = 256x^5y^8$

Now, multiply these two results together:

$1287 \times 256x^5y^8 = 329472x^5y^8$

So, the ninth term in the binomial expansion of $(x - 2y)^{13}$ is $329,472x^5y^8$.

Checking Our Answer

It’s always a good idea to double-check your answer, especially in math problems. You can use a calculator or an online binomial expansion calculator to verify your result. This step is like putting a safety net under your work – it gives you confidence that you've got the right answer and catches any small errors you might have missed.

In this case, we can be confident that our calculations are correct. We carefully followed the Binomial Theorem formula, broke the problem down into manageable steps, and double-checked our arithmetic along the way. This is the approach that will lead you to success in binomial expansion problems and many other areas of math.

Conclusion

So there you have it! The ninth term in the binomial expansion of $(x - 2y)^{13}$ is $329,472x^5y^8$. We nailed it! By understanding the Binomial Theorem and breaking down the problem into smaller, manageable steps, you can tackle even the most intimidating-looking binomial expansions. Remember the formula, practice regularly, and you'll be a binomial expansion master in no time. You guys got this!

Final Answer

The correct answer is A. $329,472 x^5 y^8$.