Simplify, For Values Of $x$ Where All Expressions Are Defined:$\[ \frac{4x^2}{x^2-4} \cdot \frac{x-2}{x^3} \\]Answer:Numerator: $\square$Denominator: $\square$

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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 focus on simplifying the given expression 4x2x2−4⋅x−2x3\frac{4x^2}{x^2-4} \cdot \frac{x-2}{x^3}, where all expressions are defined. We will break down the process into manageable steps, making it easier to understand and follow along.

Understanding the Expression

Before we dive into simplifying the expression, let's take a closer look at what we're dealing with. The given expression is a product of two fractions:

4x2x2−4⋅x−2x3\frac{4x^2}{x^2-4} \cdot \frac{x-2}{x^3}

We can see that both fractions have a common factor of x2x^2 in the numerator and a difference of squares in the denominator. Our goal is to simplify this expression by canceling out any common factors and combining like terms.

Step 1: Factor the Denominator

The first step in simplifying the expression is to factor the denominator of the first fraction. We can use the difference of squares formula to factor x2−4x^2-4:

x2−4=(x+2)(x−2)x^2-4 = (x+2)(x-2)

Now that we have factored the denominator, we can rewrite the expression as:

4x2(x+2)(x−2)⋅x−2x3\frac{4x^2}{(x+2)(x-2)} \cdot \frac{x-2}{x^3}

Step 2: Cancel Common Factors

The next step is to cancel out any common factors between the numerator and denominator. We can see that both the numerator and denominator have a factor of x−2x-2. We can cancel this factor out:

4x2(x+2)(x−2)⋅x−2x3=4x2(x+2)(x−2)⋅1x3\frac{4x^2}{(x+2)(x-2)} \cdot \frac{x-2}{x^3} = \frac{4x^2}{(x+2)(x-2)} \cdot \frac{1}{x^3}

Step 3: Simplify the Expression

Now that we have canceled out the common factor, we can simplify the expression further. We can start by canceling out any common factors between the numerator and denominator. In this case, we can cancel out a factor of x2x^2:

4x2(x+2)(x−2)⋅1x3=4(x+2)(x−2)⋅1x\frac{4x^2}{(x+2)(x-2)} \cdot \frac{1}{x^3} = \frac{4}{(x+2)(x-2)} \cdot \frac{1}{x}

Step 4: Combine Like Terms

The final step is to combine like terms. We can see that the numerator and denominator have a common factor of xx. We can cancel this factor out:

4(x+2)(x−2)⋅1x=4(x+2)(x−2)x\frac{4}{(x+2)(x-2)} \cdot \frac{1}{x} = \frac{4}{(x+2)(x-2)x}

Conclusion

In conclusion, we have successfully simplified the given expression 4x2x2−4⋅x−2x3\frac{4x^2}{x^2-4} \cdot \frac{x-2}{x^3} by canceling out common factors and combining like terms. The simplified expression is:

4(x+2)(x−2)x\frac{4}{(x+2)(x-2)x}

This expression is now in its simplest form, and we can use it to solve equations and make predictions about the behavior of the function.

Discussion

Simplifying algebraic expressions is an essential skill for any math enthusiast. By following the steps outlined in this article, you can simplify even the most complex expressions and make predictions about the behavior of the function. Remember to always factor the denominator, cancel common factors, and combine like terms to simplify the expression.

Final Answer

The final answer is:

Numerator: 4\boxed{4} Denominator: (x+2)(x−2)x\boxed{(x+2)(x-2)x}

Note: The final answer is in the format specified in the discussion category.

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Introduction

In our previous article, we explored the process of simplifying algebraic expressions, focusing on the given expression 4x2x2−4⋅x−2x3\frac{4x^2}{x^2-4} \cdot \frac{x-2}{x^3}. We broke down the process into manageable steps, making it easier to understand and follow along. In this article, we will address some common questions and concerns that may arise when simplifying algebraic expressions.

Q&A

Q: What is the first step in simplifying an algebraic expression?

A: The first step in simplifying an algebraic expression is to factor the denominator, if possible. This will help you identify any common factors that can be canceled out.

Q: How do I know if a factor can be canceled out?

A: To determine if a factor can be canceled out, look for common factors between the numerator and denominator. If you find a common factor, you can cancel it out.

Q: What is the difference of squares formula?

A: The difference of squares formula is a2−b2=(a+b)(a−b)a^2 - b^2 = (a+b)(a-b). This formula can be used to factor expressions of the form x2−4x^2 - 4, where x2x^2 is the square of a binomial.

Q: Can I cancel out a factor that is not common to both the numerator and denominator?

A: No, you cannot cancel out a factor that is not common to both the numerator and denominator. Canceling out a factor that is not common can lead to an incorrect result.

Q: How do I know if an expression is in its simplest form?

A: An expression is in its simplest form when there are no common factors that can be canceled out. You can check if an expression is in its simplest form by factoring the denominator and looking for common factors.

Q: What is the final step in simplifying an algebraic expression?

A: The final step in simplifying an algebraic expression is to combine like terms. This involves adding or subtracting terms that have the same variable and exponent.

Common Mistakes

Mistake 1: Canceling out a factor that is not common to both the numerator and denominator

Canceling out a factor that is not common can lead to an incorrect result. Make sure to only cancel out factors that are common to both the numerator and denominator.

Mistake 2: Not factoring the denominator

Failing to factor the denominator can make it difficult to identify common factors and cancel them out. Always factor the denominator, if possible.

Mistake 3: Not combining like terms

Failing to combine like terms can result in an expression that is not in its simplest form. Make sure to combine like terms after canceling out common factors.

Conclusion

Simplifying algebraic expressions can be a challenging task, but with practice and patience, you can become proficient in simplifying even the most complex expressions. Remember to always factor the denominator, cancel common factors, and combine like terms to simplify the expression. By following these steps and avoiding common mistakes, you can ensure that your expressions are in their simplest form.

Final Tips

  • Always factor the denominator, if possible.
  • Cancel common factors between the numerator and denominator.
  • Combine like terms after canceling out common factors.
  • Check if an expression is in its simplest form by factoring the denominator and looking for common factors.

By following these tips and avoiding common mistakes, you can become a master of simplifying algebraic expressions.