1. Which Is Equivalent To $(3x-4)^4$?A. $x^4+4x^2y+6x^2y^2+4xy^2+y^4$B. \$(3x-4)(3x-4)(3x-4)(3x-4)$[/tex\]C. $9x^2-24x+16$D. $(3x+4)(3x+4)(3x-4)(3x-4)$

1.1 Introduction to Exponents and Binomial Expansion

When dealing with expressions involving exponents, it's essential to understand the rules of exponentiation and how to apply them to simplify complex expressions. One of the most powerful tools in algebra is the binomial theorem, which allows us to expand expressions of the form $(a+b)^n$ into a sum of terms. In this case, we're given the expression $(3x-4)^4$ and asked to find an equivalent expression.

1.2 Understanding the Binomial Theorem

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

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

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

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

1.3 Applying the Binomial Theorem to the Given Expression

To find an equivalent expression for $(3x-4)^4$, we can apply the binomial theorem with $a=3x$ and $b=-4$. This gives us:

$$(3x-4)^4 = \binom{4}{0} (3x)^4 (-4)^0 + \binom{4}{1} (3x)^3 (-4)^1 + \binom{4}{2} (3x)^2 (-4)^2 + \binom{4}{3} (3x)^1 (-4)^3 + \binom{4}{4} (3x)^0 (-4)^4$$

1.4 Simplifying the Expression

Now, let's simplify each term in the expression:

$$\binom{4}{0} (3x)^4 (-4)^0 = 1 \cdot (3x)^4 \cdot 1 = 81x^4$$

$$\binom{4}{1} (3x)^3 (-4)^1 = 4 \cdot (3x)^3 \cdot (-4) = -432x^3$$

$$\binom{4}{2} (3x)^2 (-4)^2 = 6 \cdot (3x)^2 \cdot 16 = 864x^2$$

$$\binom{4}{3} (3x)^1 (-4)^3 = 4 \cdot (3x) \cdot (-64) = -768x$$

$$\binom{4}{4} (3x)^0 (-4)^4 = 1 \cdot 1 \cdot 256 = 256$$

1.5 Combining the Terms

Now, let's combine the terms to get the final expression:

$$(3x-4)^4 = 81x^4 - 432x^3 + 864x^2 - 768x + 256$$

1.6 Conclusion

In conclusion, the equivalent expression for $(3x-4)^4$ is $81x^4 - 432x^3 + 864x^2 - 768x + 256$. This expression can be obtained by applying the binomial theorem to the given expression and simplifying the resulting terms.

1.7 Comparison with the Given Options

Now, let's compare the obtained expression with the given options:

A. $x^4+4x2y+6x^2y2+4xy^2+y4$

B. $(3x-4)(3x-4)(3x-4)(3x-4)$

C. $9x^2-24x+16$

D. $(3x+4)(3x+4)(3x-4)(3x-4)$

It's clear that option C is not equivalent to the obtained expression. Option D is also not equivalent, as it contains the term $(3x+4)$, which is not present in the obtained expression. Option A is also not equivalent, as it contains the term $y$, which is not present in the obtained expression. Therefore, the correct answer is option B.

1.8 Final Answer

The final answer is B. $(3x-4)(3x-4)(3x-4)(3x-4)$

2.1 Introduction to Q&A

In the previous section, we explored the concept of exponents and binomial expansion, and applied it to find an equivalent expression for $(3x-4)^4$. In this section, we'll answer some frequently asked questions related to exponents and binomial expansion.

2.2 Q&A: What is the binomial theorem?

Q: What is the binomial theorem? A: The binomial theorem is a mathematical formula that allows us to expand expressions of the form $(a+b)^n$ into a sum of terms.

2.3 Q&A: How do I apply the binomial theorem?

Q: How do I apply the binomial theorem to an expression? A: To apply the binomial theorem, you need to identify the values of $a$ and $b$ in the expression, and then use the formula to expand it.

2.4 Q&A: What is the difference between a binomial and a polynomial?

Q: What is the difference between a binomial and a polynomial? A: A binomial is an expression of the form $(a+b)^n$, while a polynomial is an expression of the form $a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0$.

2.5 Q&A: How do I simplify expressions involving exponents?

Q: How do I simplify expressions involving exponents? A: To simplify expressions involving exponents, you need to apply the rules of exponentiation, such as the product rule and the power rule.

2.6 Q&A: What is the significance of the binomial coefficient?

Q: What is the significance of the binomial coefficient? A: The binomial coefficient is a number that appears in the binomial theorem, and it represents the number of ways to choose $k$ items from a set of $n$ items.

2.7 Q&A: How do I use the binomial theorem to solve problems?

Q: How do I use the binomial theorem to solve problems? A: To use the binomial theorem to solve problems, you need to identify the values of $a$ and $b$ in the expression, and then use the formula to expand it. You can then simplify the resulting expression to find the solution.

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

Q: What are some common applications of the binomial theorem? A: The binomial theorem has many applications in mathematics, science, and engineering, including the calculation of probabilities, the solution of differential equations, and the analysis of data.

2.9 Q&A: How do I choose the correct option when applying the binomial theorem?

Q: How do I choose the correct option when applying the binomial theorem? A: To choose the correct option, you need to carefully read the problem and identify the values of $a$ and $b$ in the expression. You can then use the binomial theorem to expand the expression and simplify it to find the solution.

2.10 Q&A: What are some common mistakes to avoid when applying the binomial theorem?

Q: What are some common mistakes to avoid when applying the binomial theorem? A: Some common mistakes to avoid when applying the binomial theorem include:

• Not identifying the values of $a$ and $b$ in the expression
• Not using the correct formula to expand the expression
• Not simplifying the resulting expression to find the solution
• Not checking the solution for errors

2.11 Conclusion

In conclusion, the binomial theorem is a powerful tool in mathematics that allows us to expand expressions of the form $(a+b)^n$ into a sum of terms. By understanding the binomial theorem and applying it correctly, we can solve a wide range of problems in mathematics, science, and engineering.

2.12 Final Answer

The final answer is B. $(3x-4)(3x-4)(3x-4)(3x-4)$