Select The Expression That Is Equivalent To \left(81 X^2 Y^6 - 49 Z\right ].1. \left(9 X Y^4 - 7 Z\right)\left(9 X Y^4 + 7 Z\right ]2. \left(9 X Y^3 - 7 Z\right)\left(9 X Y^3 + 7 Z\right ]3. $81 X^2 Y^8 + 63 X Y^4 Z +

Introduction

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is an essential skill for any math enthusiast. In this article, we will explore the process of simplifying algebraic expressions, focusing on the given expression $\left(81 x^2 y^6 - 49 z\right)$. We will examine three possible equivalent expressions and determine which one is correct.

Understanding the Given Expression

The given expression is $\left(81 x^2 y^6 - 49 z\right)$. To simplify this expression, we need to identify any common factors or patterns that can be used to rewrite it in a more manageable form.

Factoring the Expression

One approach to simplifying the expression is to factor it using the difference of squares formula. The difference of squares formula states that $a^2 - b^2 = (a + b)(a - b)$.

\left(81 x^2 y^6 - 49 z\right) = \left(9 x y^3\right)^2 - \left(7 z\right)^2

Using the difference of squares formula, we can rewrite the expression as:

\left(81 x^2 y^6 - 49 z\right) = \left(9 x y^3 + 7 z\right)\left(9 x y^3 - 7 z\right)

Evaluating the Options

Now that we have factored the expression, we can evaluate the three options provided:

Option 1: $\left(9 x y^4 - 7 z\right)\left(9 x y^4 + 7 z\right)$

This option is not equivalent to the factored expression. The first term in the first parentheses is $9 x y^4$, which is not the same as $9 x y^3$ in the factored expression.

Option 2: $\left(9 x y^3 - 7 z\right)\left(9 x y^3 + 7 z\right)$

This option is equivalent to the factored expression. The first term in the first parentheses is $9 x y^3$, which matches the factored expression.

Option 3: $81 x^2 y^8 + 63 x y^4 z$

This option is not equivalent to the factored expression. The expression is not in the same form as the factored expression, and the terms are not the same.

Conclusion

Based on our analysis, the correct equivalent expression is $\left(9 x y^3 - 7 z\right)\left(9 x y^3 + 7 z\right)$. This expression is the result of factoring the given expression using the difference of squares formula.

Tips and Tricks

When simplifying algebraic expressions, it's essential to identify common factors or patterns that can be used to rewrite the expression in a more manageable form. In this case, we used the difference of squares formula to factor the expression.

Common Mistakes to Avoid

When simplifying algebraic expressions, it's easy to make mistakes. Here are some common mistakes to avoid:

• Not identifying common factors or patterns
• Not using the correct formula or technique
• Not checking the work for errors

Real-World Applications

Simplifying algebraic expressions has many real-world applications. For example, in physics, algebraic expressions are used to describe the motion of objects. In engineering, algebraic expressions are used to design and optimize systems.

Practice Problems

To practice simplifying algebraic expressions, try the following problems:

• Simplify the expression $\left(16 x^2 y^4 - 25 z^2\right)$.
• Simplify the expression $\left(9 x^2 y^2 - 16 z^2\right)$.

Conclusion

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Q: What is the difference of squares formula?

A: The difference of squares formula is a mathematical formula that states $a^2 - b^2 = (a + b)(a - b)$. This formula can be used to simplify expressions that are in the form of a difference of squares.

Q: How do I identify common factors or patterns in an algebraic expression?

A: To identify common factors or patterns in an algebraic expression, look for terms that have a common factor or a common pattern. For example, if an expression has two terms that are both perfect squares, you can use the difference of squares formula to simplify it.

Q: What is the importance of simplifying algebraic expressions?

A: Simplifying algebraic expressions is important because it can help to make complex expressions more manageable and easier to work with. It can also help to identify patterns and relationships between variables.

Q: How do I know which formula or technique to use when simplifying an algebraic expression?

A: To determine which formula or technique to use when simplifying an algebraic expression, look at the expression and identify any common factors or patterns. Then, use the appropriate formula or technique to simplify the expression.

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

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

• Not identifying common factors or patterns
• Not using the correct formula or technique
• Not checking the work for errors

Q: How do I check my work when simplifying an algebraic expression?

A: To check your work when simplifying an algebraic expression, plug the simplified expression back into the original expression and verify that it is true. You can also use a calculator or computer software to check your work.

Q: What are some real-world applications of simplifying algebraic expressions?

A: Simplifying algebraic expressions has many real-world applications, including:

• Physics: Algebraic expressions are used to describe the motion of objects.
• Engineering: Algebraic expressions are used to design and optimize systems.
• Computer Science: Algebraic expressions are used to write algorithms and programs.

Q: How do I practice simplifying algebraic expressions?

A: To practice simplifying algebraic expressions, try the following:

• Simplify expressions on your own
• Use online resources or software to practice simplifying expressions
• Work with a partner or tutor to practice simplifying expressions

Q: What are some advanced techniques for simplifying algebraic expressions?

A: Some advanced techniques for simplifying algebraic expressions include:

• Using the quadratic formula to simplify quadratic expressions
• Using the rational root theorem to simplify rational expressions
• Using the binomial theorem to simplify binomial expressions

Q: How do I know when to use the quadratic formula?

A: To use the quadratic formula, the expression must be in the form of $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants. The quadratic formula is then used to find the solutions to the equation.

Q: What are some common mistakes to avoid when using the quadratic formula?

A: Some common mistakes to avoid when using the quadratic formula include:

• Not identifying the correct values of $a$, $b$, and $c$
• Not using the correct formula or technique
• Not checking the work for errors

Q: How do I know when to use the rational root theorem?

A: To use the rational root theorem, the expression must be in the form of $a/b$, where $a$ and $b$ are integers. The rational root theorem is then used to find the rational roots of the expression.

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

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

• Not identifying the correct values of $a$ and $b$
• Not using the correct formula or technique
• Not checking the work for errors

Q: How do I know when to use the binomial theorem?

A: To use the binomial theorem, the expression must be in the form of $(a + b)^n$, where $a$ and $b$ are constants and $n$ is a positive integer. The binomial theorem is then used to expand the expression.

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 correct values of $a$, $b$, and $n$
• Not using the correct formula or technique
• Not checking the work for errors