The Binomial TheoremThe Expansion Of ( 4 X K + 5 Y 2 ) 6 \left(4 X^k + 5 Y^2\right)^6 ( 4 X K + 5 Y 2 ) 6 Contains The Term 150000 X 6 Y 8 150000 X^6 Y^8 150000 X 6 Y 8 . Determine The Value Of K K K .

Introduction

The binomial theorem is a fundamental concept in algebra that allows us to expand algebraic expressions of the form $(a + b)^n$, where $a$ and $b$ are any real numbers and $n$ is a positive integer. This theorem is a powerful tool for simplifying complex expressions and has numerous applications in various fields, including mathematics, physics, engineering, and economics. In this article, we will explore the binomial theorem and its applications, with a focus on the expansion of $\left(4 x^k + 5 y^2\right)^6$ and the determination of the value of $k$.

The Binomial Theorem

The binomial theorem states that for any positive integer $n$, the expansion of $(a + b)^n$ is given by:

$$(a + b)^n = \binom{n}{0} a^n b^0 + \binom{n}{1} a^{n-1} b^1 + \binom{n}{2} a^{n-2} b^2 + \cdots + \binom{n}{n-1} a^1 b^{n-1} + \binom{n}{n} a^0 b^n$$

where $\binom{n}{k}$ is the binomial coefficient, defined as:

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

Expansion of $\left(4 x^k + 5 y^2\right)^6$

To determine the value of $k$, we need to expand the expression $\left(4 x^k + 5 y^2\right)^6$ using the binomial theorem. The expansion of this expression is given by:

$$\left(4 x^k + 5 y^2\right)^6 = \binom{6}{0} (4 x^k)^6 (5 y^2)^0 + \binom{6}{1} (4 x^k)^5 (5 y^2)^1 + \binom{6}{2} (4 x^k)^4 (5 y^2)^2 + \cdots + \binom{6}{6} (4 x^k)^0 (5 y^2)^6$$

Simplifying the above expression, we get:

$$\left(4 x^k + 5 y^2\right)^6 = 4096 x^{6k} + 28800 x^{5k} y^2 + 60480 x^{4k} y^4 + 60480 x^{3k} y^6 + 28800 x^{2k} y^8 + 4096 y^{12}$$

Determination of the Value of $k$

We are given that the expansion of $\left(4 x^k + 5 y^2\right)^6$ contains the term $150000 x^6 y^8$. To determine the value of $k$, we need to find the term in the expansion that contains $x^6 y^8$. From the simplified expression, we can see that the term $28800 x^{2k} y^8$ contains $x^6 y^8$. Therefore, we can equate the coefficients of $x^6 y^8$ in the given term and the term in the expansion:

$$150000 = 28800 x^{2k-6}$$

Simplifying the above equation, we get:

$$\frac{150000}{28800} = x^{2k-6}$$

$$\frac{25}{4} = x^{2k-6}$$

Taking the logarithm of both sides, we get:

$$\log \left(\frac{25}{4}\right) = (2k-6) \log x$$

Simplifying the above equation, we get:

$$\frac{\log 25 - \log 4}{2k-6} = \log x$$

Solving for $k$, we get:

$$k = \frac{6 + \log 4 - \log 25}{2 \log x}$$

Conclusion

In this article, we have explored the binomial theorem and its applications, with a focus on the expansion of $\left(4 x^k + 5 y^2\right)^6$ and the determination of the value of $k$. We have shown that the binomial theorem is a powerful tool for simplifying complex expressions and has numerous applications in various fields. We have also determined the value of $k$ by equating the coefficients of $x^6 y^8$ in the given term and the term in the expansion.

References

• [1] Binomial Theorem. (n.d.). In Encyclopedia Britannica. Retrieved from https://www.britannica.com/topic/binomial-theorem
• [2] Algebra. (n.d.). In Khan Academy. Retrieved from https://www.khanacademy.org/math/algebra
• [3] Binomial Coefficient. (n.d.). In Wolfram MathWorld. Retrieved from https://mathworld.wolfram.com/BinomialCoefficient.html

Further Reading

• [1] Binomial Theorem. (n.d.). In Math Open Reference. Retrieved from https://www.mathopenref.com/binomialtheorem.html
• [2] Algebra. (n.d.). In Math Is Fun. Retrieved from https://www.mathisfun.com/algebra/
• [3] Binomial Coefficient. (n.d.). In Mathworld. Retrieved from https://mathworld.wolfram.com/BinomialCoefficient.html
  The Binomial Theorem: A Q&A Guide =====================================

Introduction

The binomial theorem is a fundamental concept in algebra that allows us to expand algebraic expressions of the form $(a + b)^n$, where $a$ and $b$ are any real numbers and $n$ is a positive integer. In our previous article, we explored the binomial theorem and its applications, with a focus on the expansion of $\left(4 x^k + 5 y^2\right)^6$ and the determination of the value of $k$. In this article, we will provide a Q&A guide to help you better understand the binomial theorem and its applications.

Q: What is the binomial theorem?

A: The binomial theorem is a mathematical formula that allows us to expand algebraic expressions of the form $(a + b)^n$, where $a$ and $b$ are any real numbers and $n$ is a positive integer.

Q: What is the formula for the binomial theorem?

A: The formula for the binomial theorem is:

$$(a + b)^n = \binom{n}{0} a^n b^0 + \binom{n}{1} a^{n-1} b^1 + \binom{n}{2} a^{n-2} b^2 + \cdots + \binom{n}{n-1} a^1 b^{n-1} + \binom{n}{n} a^0 b^n$$

Q: What is the binomial coefficient?

A: The binomial coefficient is a number that appears in the formula for the binomial theorem. It is defined as:

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

Q: How do I use the binomial theorem to expand an algebraic expression?

A: To use the binomial theorem to expand an algebraic expression, you need to follow these steps:

1. Identify the values of $a$, $b$, and $n$ in the expression.
2. Plug these values into the formula for the binomial theorem.
3. Simplify the resulting expression.

Q: What are some common applications of the binomial theorem?

A: The binomial theorem has numerous applications in various fields, including:

• Algebra: The binomial theorem is used to expand algebraic expressions and simplify complex equations.
• Calculus: The binomial theorem is used to find the derivatives and integrals of functions.
• Statistics: The binomial theorem is used to calculate probabilities and expected values.
• Physics: The binomial theorem is used to describe the motion of particles and the behavior of systems.

Q: How do I determine the value of $k$ in the expansion of $\left(4 x^k + 5 y^2\right)^6$?

A: To determine the value of $k$, you need to equate the coefficients of $x^6 y^8$ in the given term and the term in the expansion. This will give you an equation that you can solve for $k$.

Q: What are some common mistakes to avoid when using the binomial theorem?

A: Some common mistakes to avoid when using the binomial theorem include:

• Not identifying the values of $a$, $b$, and $n$ correctly.
• Not simplifying the resulting expression correctly.
• Not using the correct formula for the binomial coefficient.

Conclusion

In this article, we have provided a Q&A guide to help you better understand the binomial theorem and its applications. We have covered topics such as the formula for the binomial theorem, the binomial coefficient, and common applications of the binomial theorem. We have also provided tips and tricks for using the binomial theorem to expand algebraic expressions and determine the value of $k$.

References

• [1] Binomial Theorem. (n.d.). In Encyclopedia Britannica. Retrieved from https://www.britannica.com/topic/binomial-theorem
• [2] Algebra. (n.d.). In Khan Academy. Retrieved from https://www.khanacademy.org/math/algebra
• [3] Binomial Coefficient. (n.d.). In Wolfram MathWorld. Retrieved from https://mathworld.wolfram.com/BinomialCoefficient.html

Further Reading

• [1] Binomial Theorem. (n.d.). In Math Open Reference. Retrieved from https://www.mathopenref.com/binomialtheorem.html
• [2] Algebra. (n.d.). In Math Is Fun. Retrieved from https://www.mathisfun.com/algebra/
• [3] Binomial Coefficient. (n.d.). In Mathworld. Retrieved from https://mathworld.wolfram.com/BinomialCoefficient.html