Finding The Sixth Term In Binomial Expansion Of (2a - 3b)^10

Hey guys! Let's dive into the binomial expansion and figure out which expression represents the sixth term in the expansion of $(2a - 3b)^{10}$. This might seem daunting at first, but trust me, we'll break it down step by step so it's super easy to understand. Binomial expansion is a powerful tool in algebra, and mastering it can really boost your math skills. So, let’s get started and tackle this problem together!

What is Binomial Expansion?

Before we jump into the specifics of the question, let’s quickly recap what binomial expansion actually means. A binomial is simply an algebraic expression with two terms, like $(x + y)$ or, in our case, $(2a - 3b)$. When we raise a binomial to a power, we’re essentially multiplying it by itself multiple times. For example, $(x + y)^2$ means $(x + y)(x + y)$, and expanding this gives us $x^2 + 2xy + y^2$. Now, imagine doing this for something like $(2a - 3b)^{10}$ – that’s where things can get a little hairy! Manually multiplying this out would take ages and be prone to errors. Thankfully, there’s a much smarter way: the Binomial Theorem.

The Binomial Theorem provides a formula to expand binomials raised to any power without actually doing all the multiplication. The general form of the Binomial Theorem is:

$(a + b)^n = \sum_{k=0}^{n} {n \choose k} a^{n-k} b^k$

Where:

• $n$ is the power to which the binomial is raised.
• $k$ is an index that ranges from 0 to $n$.
• ${n \choose k}$ is the binomial coefficient, often read as "n choose k," and it represents the number of ways to choose $k$ items from a set of $n$ items. It's calculated as ${n \choose k} = \frac{n!}{k!(n-k)!}$, where $!$ denotes the factorial (e.g., $5! = 5 \times 4 \times 3 \times 2 \times 1$).

So, the Binomial Theorem tells us exactly how each term in the expansion looks. The coefficients ${n \choose k}$ give us the numerical parts, and the powers of $a$ and $b$ follow a predictable pattern. The exponent of $a$ starts at $n$ and decreases by 1 with each term, while the exponent of $b$ starts at 0 and increases by 1 with each term. Understanding this theorem is crucial for finding specific terms in the expansion, like the sixth term we’re after.

In essence, the Binomial Theorem is a shortcut that saves us a ton of time and effort. Instead of manually multiplying the binomial by itself ten times, we can use the formula to directly calculate any term in the expansion. This is especially useful when dealing with large exponents, making problems like this one much more manageable. Keep in mind that each term in the expansion corresponds to a specific value of $k$, and we’ll need to figure out which $k$ gives us the sixth term. So, with the basics of the theorem in our toolkit, let’s move on to applying it to our problem and pinpointing the expression for the sixth term.

Identifying the Sixth Term

Okay, now that we’ve got the Binomial Theorem fresh in our minds, let's figure out how to apply it to our specific problem: finding the sixth term in the expansion of $(2a - 3b)^{10}$. The key here is understanding how the terms are numbered in the expansion. Remember that the index $k$ in the Binomial Theorem starts at 0, not 1. This means the first term corresponds to $k = 0$, the second term corresponds to $k = 1$, and so on. So, to find the sixth term, we need to look at the term where $k = 5$ (since $0, 1, 2, 3, 4, 5$ gives us six terms).

Using the general formula from the Binomial Theorem, we can write the term for $k = 5$ as:

${10 \choose 5} (2a)^{10-5} (-3b)^5$

Here's how we plugged in the values:

• $n = 10$ (the exponent of the binomial)
• $k = 5$ (since we want the sixth term)
• $a = 2a$ (the first term in the binomial)
• $b = -3b$ (the second term in the binomial)

Now, let's simplify this expression a bit. We have:

${10 \choose 5} (2a)^5 (-3b)^5$

Let's break this down further. First, we need to remember that the binomial coefficient ${10 \choose 5}$ is calculated as:

${10 \choose 5} = \frac{10!}{5!(10-5)!} = \frac{10!}{5!5!}$

While we could calculate this value directly, for the purpose of this question, we're mainly interested in the form of the expression, not the numerical result just yet. Next, we have $(2a)^5$, which means $2^5 \times a^5 = 32a^5$. And finally, $(-3b)^5$ means $(-3)^5 \times b^5 = -243b^5$. The negative sign here is crucial because it alternates the signs of the terms in the expansion due to the negative term in the original binomial.

So, putting it all together, the sixth term looks like:

${10 \choose 5} (32a^5) (-243b^5)$

${10 \choose 5} (2a)^5 (-3b)^5$

This expression matches one of the options provided, so we're on the right track! Remember, the key to finding specific terms in binomial expansions is to correctly identify the value of $k$ and plug it into the Binomial Theorem formula. And don't forget about the negative signs – they can make a big difference! Now, let’s take a look at the answer choices and see which one matches our result.

Matching the Expression with the Answer Choices

Alright, we've successfully derived the expression for the sixth term in the binomial expansion of $(2a - 3b)^{10}$. Our expression is:

${10 \choose 5} (2a)^5 (-3b)^5$

Now, let's compare this with the answer choices provided in the question. This is where we need to be a bit meticulous and make sure we don't miss any details, especially the signs and the exponents.

Looking at the options, we need to find one that has:

• The binomial coefficient ${10 \choose 5}$
• $(2a)$ raised to the power of 5
• $(-3b)$ raised to the power of 5

Let’s go through some of the options to see if they match:

• Option 1: ${10 \choose 5} (2a)^5 (-3b)^5$
  • This option looks promising! It has the correct binomial coefficient, $(2a)^5$, and $(-3b)^5$. This seems to be a direct match to what we derived. So, we might have found our answer right away!
• Option 2: ${10 \choose 5} (-2a)^5 (3b)^5$
  • This option has the correct binomial coefficient and exponents, but the signs are different. Notice that $(-2a)^5$ would result in a negative term, and $(3b)^5$ is positive. This means the whole term would be negative. While the magnitude might be correct, the sign is different from our derived expression, where we have $(-3b)^5$, which is also negative, but the $2a$ term is positive. So, this is not the correct answer.
• Option 3: ${10 \choose 6} (2a)^4 (-3b)^6$
  • This option has a different binomial coefficient, ${10 \choose 6}$, and different exponents for $(2a)$ and $(-3b)$. Remember, we identified that for the sixth term, we need to use $k = 5$, which gives us ${10 \choose 5}$. Also, the powers of $(2a)$ and $(-3b)$ should be 5 based on our calculations. So, this option is incorrect.
• Option 4: ${10 \choose 6} (-2a)^4 (3b)^6$
  • Similar to Option 3, this has the wrong binomial coefficient and exponents. Additionally, even if the exponents were correct, $(-2a)^4$ would be positive because any negative number raised to an even power is positive, and $(3b)^6$ is also positive. This would result in a positive term, which doesn't match the sign we expect.
• Option 5: ${10 \choose 6} (2a)^6 (-3b)^4$
  • This option also has the incorrect binomial coefficient and exponents. The power of $(2a)$ should be 5, and the power of $(-3b)$ should also be 5. So, this option is not the correct one.

By carefully comparing each option with our derived expression, we can confidently say that Option 1: ${10 \choose 5} (2a)^5 (-3b)^5$ is the correct answer. This process highlights the importance of paying attention to every detail in the expression, especially the coefficients, exponents, and signs. Getting any of these wrong would lead to an incorrect answer. So, always double-check your work and make sure everything lines up perfectly!

Final Answer

Okay, we've worked through the entire problem step by step, and it's time to state our final answer! We started by understanding the Binomial Theorem, figured out how to identify the sixth term in the expansion of $(2a - 3b)^{10}$, and then carefully compared our derived expression with the answer choices. After a thorough analysis, we pinpointed the correct answer.

The expression that represents the sixth term in the binomial expansion of $(2a - 3b)^{10}$ is:

${10 \choose 5} (2a)^5 (-3b)^5$

This corresponds to the first option provided in the question. Remember, the key takeaways from this problem are:

1. Understanding the Binomial Theorem is crucial for expanding binomials efficiently.
2. The index $k$ starts at 0, so the sixth term corresponds to $k = 5$.
3. Pay close attention to signs and exponents when plugging values into the formula.
4. Carefully compare your derived expression with the answer choices to avoid errors.

By mastering these concepts, you'll be well-equipped to tackle similar problems involving binomial expansion. So, keep practicing, and you'll become a pro at these types of questions in no time! Great job, guys! We nailed it!