Rewrite The Expression: \[$(x - 7y) \cdot (x - 7y) \cdot (x - 7y) \cdot (x - 7y)\$\]

Introduction

In this article, we will delve into the world of algebra and explore the process of simplifying a given expression. The expression we will be working with is {(x - 7y) \cdot (x - 7y) \cdot (x - 7y) \cdot (x - 7y)$}$. This expression involves the multiplication of four identical binomials, and our goal is to simplify it to its most basic form.

Understanding the Expression

Before we begin simplifying the expression, let's take a closer look at what we're dealing with. The expression consists of four identical binomials, each in the form of ${(x - 7y)}$. When we multiply these binomials together, we will need to apply the distributive property and combine like terms.

Step 1: Apply the Distributive Property

To simplify the expression, we will start by applying the distributive property. This property states that for any real numbers ${a}$, ${b}$, and ${c}$, the following equation holds: ${a \cdot (b + c) = a \cdot b + a \cdot c}$. We will use this property to expand the expression and combine like terms.

(x - 7y) \cdot (x - 7y) \cdot (x - 7y) \cdot (x - 7y) = (x - 7y)^4

Step 2: Expand the Expression

Now that we have applied the distributive property, we can expand the expression. To do this, we will use the binomial theorem, which states that for any positive integer ${n}$, the following equation holds: ${(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k}$. In our case, we have ${(x - 7y)^4}$, so we will use the binomial theorem to expand it.

(x - 7y)^4 = \binom{4}{0} x^4 + \binom{4}{1} x^3 (-7y) + \binom{4}{2} x^2 (-7y)^2 + \binom{4}{3} x (-7y)^3 + \binom{4}{4} (-7y)^4

Step 3: Simplify the Expression

Now that we have expanded the expression, we can simplify it by combining like terms. We will start by simplifying the coefficients of each term.

(x - 7y)^4 = x^4 - 28x^3y + 196x^2y^2 - 784xy^3 + 2401y^4

Conclusion

In this article, we have simplified the expression {(x - 7y) \cdot (x - 7y) \cdot (x - 7y) \cdot (x - 7y)$}$ using the distributive property and the binomial theorem. We have expanded the expression and combined like terms to arrive at the simplified form: {x^4 - 28x^3y + 196x^2y2 - 784xy^3 + 2401y^4}. This simplified expression is the most basic form of the original expression, and it can be used in a variety of mathematical applications.

Further Reading

If you are interested in learning more about algebra and simplifying expressions, we recommend checking out the following resources:

• Algebraic Expressions
• Binomial Theorem
• Distributive Property

Glossary

• Binomial: An algebraic expression consisting of two terms.
• Distributive Property: A property of algebra that states that for any real numbers ${a}$, ${b}$, and ${c}$, the following equation holds: ${a \cdot (b + c) = a \cdot b + a \cdot c}$.
• Binomial Theorem: A theorem that states that for any positive integer ${n}$, the following equation holds: ${(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k}$.
  Simplifying the Expression: A Q&A Guide =====================================================

Introduction

In our previous article, we explored the process of simplifying the expression {(x - 7y) \cdot (x - 7y) \cdot (x - 7y) \cdot (x - 7y)$}$. We applied the distributive property and the binomial theorem to arrive at the simplified form: {x^4 - 28x^3y + 196x^2y2 - 784xy^3 + 2401y^4}. In this article, we will answer some frequently asked questions about simplifying expressions and provide additional guidance on how to approach these types of problems.

Q&A

Q: What is the distributive property, and how is it used in simplifying expressions?

A: The distributive property is a fundamental concept in algebra that states that for any real numbers ${a}$, ${b}$, and ${c}$, the following equation holds: ${a \cdot (b + c) = a \cdot b + a \cdot c}$. This property is used to expand expressions and combine like terms.

Q: How do I apply the distributive property to simplify an expression?

A: To apply the distributive property, you need to multiply each term in the first expression by each term in the second expression. For example, if you have the expression ${(x + 3) \cdot (x + 4)}$, you would multiply each term in the first expression by each term in the second expression: ${(x + 3) \cdot (x + 4) = x \cdot x + x \cdot 4 + 3 \cdot x + 3 \cdot 4}$.

Q: What is the binomial theorem, and how is it used in simplifying expressions?

A: The binomial theorem is a mathematical formula that describes the expansion of a binomial raised to a power. It states that for any positive integer ${n}$, the following equation holds: ${(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k}$. This theorem is used to expand expressions and combine like terms.

Q: How do I apply the binomial theorem to simplify an expression?

A: To apply the binomial theorem, you need to identify the binomial and the power to which it is raised. For example, if you have the expression ${(x + 3)^4}$, you would use the binomial theorem to expand it: ${(x + 3)^4 = \binom{4}{0} x^4 + \binom{4}{1} x^3 (3) + \binom{4}{2} x^2 (3)^2 + \binom{4}{3} x (3)^3 + \binom{4}{4} (3)^4}$.

Q: What are some common mistakes to avoid when simplifying expressions?

A: Some common mistakes to avoid when simplifying expressions include:

• Not applying the distributive property correctly
• Not combining like terms
• Not using the binomial theorem when necessary
• Not checking for errors in the final answer

Q: How can I practice simplifying expressions?

A: You can practice simplifying expressions by working through examples and exercises in your textbook or online resources. You can also try simplifying expressions on your own and then checking your answers with a calculator or online tool.

Conclusion

Simplifying expressions is an essential skill in algebra and mathematics. By understanding the distributive property and the binomial theorem, you can simplify expressions and arrive at the correct answer. Remember to apply the distributive property correctly, combine like terms, and use the binomial theorem when necessary. With practice and patience, you can become proficient in simplifying expressions and tackle even the most challenging problems.

Further Reading

If you are interested in learning more about algebra and simplifying expressions, we recommend checking out the following resources:

• Algebraic Expressions
• Binomial Theorem
• Distributive Property

Glossary

• Binomial: An algebraic expression consisting of two terms.
• Distributive Property: A property of algebra that states that for any real numbers ${a}$, ${b}$, and ${c}$, the following equation holds: ${a \cdot (b + c) = a \cdot b + a \cdot c}$.
• Binomial Theorem: A theorem that states that for any positive integer ${n}$, the following equation holds: ${(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k}$.