Simplify The Expression:$\[ \frac{x^2-4}{x+3} \div \frac{4x-8}{3x+9} \\]

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Introduction

In mathematics, simplifying expressions is a crucial step in solving problems and understanding complex concepts. When dealing with fractions, division, and algebraic expressions, simplification can make a significant difference in the outcome. In this article, we will focus on simplifying the given expression: $\frac{x^2-4}{x+3} \div \frac{4x-8}{3x+9}$. We will break down the steps involved in simplifying this expression and provide a clear understanding of the process.

Understanding the Expression

The given expression involves two fractions: $\frac{x^2-4}{x+3}$ and $\frac{4x-8}{3x+9}$. The expression is in the form of a division, where the first fraction is divided by the second fraction. To simplify this expression, we need to follow the order of operations (PEMDAS):

1. Parentheses: Evaluate expressions inside parentheses.
2. Exponents: Evaluate any exponential expressions.
3. Multiplication and Division: Evaluate multiplication and division operations from left to right.
4. Addition and Subtraction: Evaluate any addition and subtraction operations from left to right.

Step 1: Factor the Numerator and Denominator

To simplify the expression, we need to factor the numerator and denominator of each fraction. The numerator of the first fraction can be factored as:

$x^2 - 4 = (x + 2)(x - 2)$

The denominator of the first fraction is already factored as:

$x + 3$

The numerator of the second fraction can be factored as:

$4x - 8 = 4(x - 2)$

The denominator of the second fraction can be factored as:

$3x + 9 = 3(x + 3)$

Step 2: Rewrite the Expression with Factored Numerators and Denominators

Now that we have factored the numerators and denominators, we can rewrite the expression as:

$\frac{(x + 2)(x - 2)}{x + 3} \div \frac{4(x - 2)}{3(x + 3)}$

Step 3: Simplify the Expression

To simplify the expression, we can cancel out common factors between the numerator and denominator of each fraction. In this case, we can cancel out the $(x - 2)$ term in the numerator and denominator of the first fraction, and the $(x + 3)$ term in the numerator and denominator of the second fraction:

$\frac{x + 2}{1} \div \frac{4}{3(x + 3)}$

Step 4: Invert and Multiply

To simplify the expression further, we can invert the second fraction and multiply:

$\frac{x + 2}{1} \times \frac{3(x + 3)}{4}$

Step 5: Simplify the Expression

Now that we have inverted and multiplied, we can simplify the expression by multiplying the numerators and denominators:

$\frac{(x + 2)(3)(x + 3)}{4}$

Step 6: Simplify the Expression

Finally, we can simplify the expression by combining like terms in the numerator:

$\frac{3x^2 + 15x + 18}{4}$

Conclusion

In this article, we simplified the given expression: $\frac{x^2-4}{x+3} \div \frac{4x-8}{3x+9}$. We broke down the steps involved in simplifying this expression and provided a clear understanding of the process. By following the order of operations and factoring the numerators and denominators, we were able to simplify the expression and arrive at the final answer.

Frequently Asked Questions

• Q: What is the order of operations? A: The order of operations is PEMDAS: Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction.
• Q: How do I factor the numerator and denominator of a fraction? A: To factor the numerator and denominator of a fraction, look for common factors and group them together.
• Q: How do I simplify an expression? A: To simplify an expression, follow the order of operations and cancel out common factors between the numerator and denominator of each fraction.

Final Answer

The final answer is: $\boxed{\frac{3x^2 + 15x + 18}{4}}$

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Introduction

In our previous article, we simplified the given expression: $\frac{x^2-4}{x+3} \div \frac{4x-8}{3x+9}$. We broke down the steps involved in simplifying this expression and provided a clear understanding of the process. In this article, we will answer some frequently asked questions related to simplifying expressions and provide additional information to help you better understand the concept.

Q&A

Q: What is the order of operations?

A: The order of operations is PEMDAS: Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction. This means that you should evaluate expressions inside parentheses first, followed by exponents, then multiplication and division from left to right, and finally addition and subtraction from left to right.

Q: How do I factor the numerator and denominator of a fraction?

A: To factor the numerator and denominator of a fraction, look for common factors and group them together. For example, if you have the fraction $\frac{x^2-4}{x+3}$, you can factor the numerator as $(x+2)(x-2)$ and the denominator as $(x+3)$.

Q: How do I simplify an expression?

A: To simplify an expression, follow the order of operations and cancel out common factors between the numerator and denominator of each fraction. For example, if you have the expression $\frac{(x+2)(x-2)}{x+3} \div \frac{4(x-2)}{3(x+3)}$, you can cancel out the $(x-2)$ term in the numerator and denominator of the first fraction, and the $(x+3)$ term in the numerator and denominator of the second fraction.

Q: What is the difference between simplifying an expression and solving an equation?

A: Simplifying an expression involves reducing the complexity of the expression by canceling out common factors and combining like terms. Solving an equation, on the other hand, involves finding the value of the variable that makes the equation true. For example, if you have the equation $x^2-4 = 0$, you can simplify the expression on the left-hand side by factoring it as $(x+2)(x-2) = 0$, but you still need to solve for $x$ to find the value of the variable.

Q: How do I know when to simplify an expression?

A: You should simplify an expression whenever it is necessary to make the expression easier to work with. For example, if you have a complex expression that involves multiple fractions and variables, simplifying it can make it easier to solve for the variable or evaluate the expression.

Q: Can I simplify an expression that involves variables with exponents?

A: Yes, you can simplify an expression that involves variables with exponents. For example, if you have the expression $\frac{x^2-4}{x+3} \div \frac{4x^2-8}{3x^2+9}$, you can simplify it by canceling out common factors and combining like terms.

Q: How do I know when to use the distributive property to simplify an expression?

A: You should use the distributive property to simplify an expression when it involves multiplying a single term by multiple terms. For example, if you have the expression $(x+2)(x-2)$, you can use the distributive property to simplify it as $x^2-2x+2x-4$.

Conclusion

In this article, we answered some frequently asked questions related to simplifying expressions and provided additional information to help you better understand the concept. We hope that this article has been helpful in clarifying any confusion you may have had about simplifying expressions.

Final Answer

The final answer is: $\boxed{\frac{3x^2 + 15x + 18}{4}}$

Additional Resources

• For more information on simplifying expressions, see our previous article: [Simplify the Expression: $\frac{x^2-4}{x+3} \div \frac{4x-8}{3x+9}$](link to previous article)
• For more information on the distributive property, see our article: [The Distributive Property: A Guide to Simplifying Expressions](link to article)
• For more information on solving equations, see our article: [Solving Equations: A Guide to Finding the Value of the Variable](link to article)