Calculate The Third Term In Binomial Expansion Of (x+y)^8

Hey guys! Let's dive into a fun math problem today. We're going to explore how to find a specific term in a binomial expansion. Specifically, we're looking at the expression $(x+y)^8$, and we want to figure out the numerical value of the third term when $x=0.3$ and $y=0.7$. It might sound a little intimidating at first, but trust me, we'll break it down step-by-step so it's super easy to understand. So, grab your thinking caps, and let's get started!

The Binomial Theorem: Our Secret Weapon

To tackle this problem, our secret weapon is the Binomial Theorem. This theorem provides a formula for expanding expressions of the form $(a+b)^n$, where $n$ is a non-negative integer. The general formula looks like this:

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

Okay, I know, that might look a little scary with all the symbols, but let's break it down. The $\sum$ symbol means we're summing up a series of terms. The ${n \choose k}$ part is a binomial coefficient, which is read as "n choose k" and 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., $5! = 5 \times 4 \times 3 \times 2 \times 1$). The $a^{n-k}$ and $b^k$ parts are just the powers of $a$ and $b$ in each term. So, in simpler terms, the Binomial Theorem tells us how to expand an expression like $(x+y)^8$ into a sum of terms involving different powers of $x$ and $y$, each with a specific coefficient.

Applying the Binomial Theorem to Our Problem

In our case, we have $(x+y)^8$, so $n = 8$. We're interested in the third term. Remember, the terms are numbered starting from $k = 0$. So, the first term corresponds to $k = 0$, the second term to $k = 1$, and the third term to $k = 2$. Therefore, we need to find the term when $k = 2$. Plugging $n = 8$ and $k = 2$ into the Binomial Theorem formula, we get:

${8 \choose 2} x^{8-2} y^2 = {8 \choose 2} x^6 y^2$

Now, let's calculate the binomial coefficient ${8 \choose 2}$:

${8 \choose 2} = \frac{8!}{2!(8-2)!} = \frac{8!}{2!6!} = \frac{8 \times 7 \times 6!}{2 \times 1 \times 6!} = \frac{8 \times 7}{2} = 28$

So, the third term is $28x^6y^2$. This is a crucial step in solving our problem, as we've now isolated the specific term we need to evaluate. We've successfully navigated the initial complexities of the Binomial Theorem and are well on our way to finding our numerical answer. The breakdown of the formula and the step-by-step calculation of the binomial coefficient should make the process clear and less daunting. Now, let's move on to the next phase – plugging in the values of $x$ and $y$ and crunching the numbers!

Plugging in the Values: x = 0.3 and y = 0.7

Now that we've found the general form of the third term, $28x^6y^2$, it's time to substitute the given values of $x$ and $y$. We know that $x = 0.3$ and $y = 0.7$. So, we just need to plug these values into our expression:

$28x^6y^2 = 28(0.3)^6(0.7)^2$

This looks a bit more manageable, doesn't it? We've gone from a complex binomial expansion to a straightforward numerical calculation. But before we reach for our calculators, let's take a moment to appreciate what we've accomplished. We've successfully applied the Binomial Theorem, identified the correct term, and now we're ready to substitute the given values. This methodical approach is key to tackling math problems effectively. Now, let's get those exponents sorted out!

Calculating the Powers

Let's calculate $(0.3)^6$ and $(0.7)^2$ separately. First, $(0.3)^6$ means $0.3$ multiplied by itself six times:

$(0.3)^6 = 0.3 \times 0.3 \times 0.3 \times 0.3 \times 0.3 \times 0.3 = 0.000729$

Next, let's calculate $(0.7)^2$:

$(0.7)^2 = 0.7 \times 0.7 = 0.49$

Now we have all the pieces we need. We've calculated the binomial coefficient, and we've found the values of $(0.3)^6$ and $(0.7)^2$. The next step is to put it all together and get our final answer. Don't worry, we're almost there! The hard work is done; now it's just a matter of multiplication. This is where careful calculation is essential to ensure we arrive at the correct answer.

Putting It All Together: The Final Calculation

Now, let's substitute the calculated values back into our expression:

$28(0.3)^6(0.7)^2 = 28 \times 0.000729 \times 0.49$

We can multiply these numbers together to get the final result. It's a good idea to use a calculator for this step to ensure accuracy, especially with decimals involved.

$28 \times 0.000729 \times 0.49 = 0.01000932$

Now, looking at the answer choices, we need to find the closest value. This is a crucial step in problem-solving, as we often need to round our answers to match the given options. We've done all the hard work, and now we need to make sure we select the correct answer based on our calculations. So, let's see which option is the best fit!

Choosing the Correct Answer

Our calculated value is approximately $0.01000932$. Now, let's compare this to the given answer choices:

A. 0.010 B. 0.020 C. 0.058 D. 0.060 E. 0.068

The closest value to our result is $0.010$. Therefore, the numerical value of the third term in the expansion of $(x+y)^8$ when $x=0.3$ and $y=0.7$ is approximately $0.010$.

So, the correct answer is A. 0.010.

Recap and Key Takeaways

We've successfully navigated this problem by using the Binomial Theorem, identifying the relevant term, substituting the given values, and performing the necessary calculations. It's a great feeling when we can break down a seemingly complex problem into manageable steps, isn't it? Let's recap the key takeaways from our journey today:

1. The Binomial Theorem: This is our go-to tool for expanding expressions of the form $(a+b)^n$.
2. Identifying the Correct Term: Remember that the terms are numbered starting from $k = 0$.
3. Substituting Values: Once you have the correct term, plug in the given values carefully.
4. Calculating Accurately: Use a calculator to avoid errors when dealing with decimals and exponents.
5. Rounding and Choosing the Closest Answer: Sometimes, you'll need to round your answer to match the available options.

By keeping these points in mind, you'll be well-equipped to tackle similar problems in the future. Math can be challenging, but with a structured approach and a clear understanding of the underlying concepts, it becomes much more accessible. And remember, guys, practice makes perfect! The more you work through problems like this, the more confident and comfortable you'll become. So, keep exploring, keep learning, and most importantly, keep having fun with math! We did it!