Simplify The Expression: ${ \frac{3x}{x^2-x-12} \div \frac{12x^2}{x-4} }$

Introduction

When it comes to simplifying complex expressions, dividing fractions can be a daunting task. However, with a clear understanding of the steps involved and a bit of practice, you'll be able to simplify even the most challenging expressions with ease. In this article, we'll take a closer look at how to simplify the expression $\frac{3x}{x^2-x-12} \div \frac{12x^2}{x-4}$ using a step-by-step approach.

Understanding the Problem

Before we dive into the solution, let's take a closer look at the expression we're trying to simplify. The expression is a division of two fractions, which can be written as:

$$\frac{3x}{x^2-x-12} \div \frac{12x^2}{x-4}$$

To simplify this expression, we need to follow the order of operations (PEMDAS):

1. Parentheses: Evaluate any expressions inside parentheses.
2. Exponents: Evaluate any exponential expressions.
3. Multiplication and Division: Evaluate any 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 Denominators

The first step in simplifying the expression is to factor the denominators of both fractions. The denominator of the first fraction is $x^2-x-12$, which can be factored as:

$$x^2-x-12 = (x-4)(x+3)$$

The denominator of the second fraction is $x-4$, which is already factored.

Step 2: Rewrite the Expression with Factored Denominators

Now that we've factored the denominators, we can rewrite the expression as:

$$\frac{3x}{(x-4)(x+3)} \div \frac{12x^2}{x-4}$$

Step 3: Invert and Multiply

To divide fractions, we need to invert the second fraction and multiply. This means that we'll multiply the first fraction by the reciprocal of the second fraction:

$$\frac{3x}{(x-4)(x+3)} \cdot \frac{x-4}{12x^2}$$

Step 4: Simplify the Expression

Now that we've inverted and multiplied, we can simplify the expression by canceling out any common factors. In this case, we can cancel out the $(x-4)$ term in the numerator and denominator:

$$\frac{3x}{(x+3)} \cdot \frac{1}{12x^2}$$

Step 5: Simplify the Expression Further

We can simplify the expression further by canceling out any common factors in the numerator and denominator. In this case, we can cancel out the $3$ term in the numerator and denominator:

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

Step 6: Simplify the Expression Even Further

We can simplify the expression even further by canceling out any common factors in the numerator and denominator. In this case, we can cancel out the $x$ term in the numerator and denominator:

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

Step 7: Final Simplification

The final step is to simplify the expression by multiplying the two fractions together:

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

And that's it! We've successfully simplified the expression $\frac{3x}{x^2-x-12} \div \frac{12x^2}{x-4}$ using a step-by-step approach.

Conclusion

Simplifying complex expressions can be a challenging task, but with a clear understanding of the steps involved and a bit of practice, you'll be able to simplify even the most challenging expressions with ease. In this article, we've taken a closer look at how to simplify the expression $\frac{3x}{x^2-x-12} \div \frac{12x^2}{x-4}$ using a step-by-step approach. By following the order of operations and simplifying the expression step by step, we've arrived at the final simplified expression: $\frac{1}{4x(x+3)}$. Whether you're a student or a professional, mastering the art of simplifying complex expressions is an essential skill that will serve you well in all areas of mathematics.

Frequently Asked Questions

• Q: What is the order of operations? A: The order of operations is a set of rules that dictates the order in which mathematical operations should be performed. The order of operations is: Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction.
• Q: How do I simplify a complex expression? A: To simplify a complex expression, follow the order of operations and simplify the expression step by step. This may involve factoring the denominators, inverting and multiplying, and canceling out common factors.
• Q: What is the difference between a fraction and a division? A: A fraction is a way of expressing a part of a whole, while a division is a way of expressing a quotient. In the expression $\frac{3x}{x^2-x-12} \div \frac{12x^2}{x-4}$, the first fraction is a way of expressing a part of a whole, while the second fraction is a way of expressing a quotient.

Additional Resources

• Khan Academy: Simplifying Complex Expressions
• Mathway: Simplifying Complex Expressions
• Wolfram Alpha: Simplifying Complex Expressions

Final Thoughts

Simplifying complex expressions is an essential skill that will serve you well in all areas of mathematics. By following the order of operations and simplifying the expression step by step, you'll be able to simplify even the most challenging expressions with ease. Whether you're a student or a professional, mastering the art of simplifying complex expressions is an essential skill that will serve you well in all areas of mathematics.

Introduction

In our previous article, we took a closer look at how to simplify the expression $\frac{3x}{x^2-x-12} \div \frac{12x^2}{x-4}$ using a step-by-step approach. However, we know that simplifying complex expressions can be a challenging task, and many of you may have questions about the process. In this article, we'll answer some of the most frequently asked questions about simplifying complex expressions, including dividing fractions.

Q&A: Simplifying Complex Expressions

Q: What is the order of operations?

A: The order of operations is a set of rules that dictates the order in which mathematical operations should be performed. The order of operations is: Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction.

Q: How do I simplify a complex expression?

A: To simplify a complex expression, follow the order of operations and simplify the expression step by step. This may involve factoring the denominators, inverting and multiplying, and canceling out common factors.

Q: What is the difference between a fraction and a division?

A: A fraction is a way of expressing a part of a whole, while a division is a way of expressing a quotient. In the expression $\frac{3x}{x^2-x-12} \div \frac{12x^2}{x-4}$, the first fraction is a way of expressing a part of a whole, while the second fraction is a way of expressing a quotient.

Q: How do I invert and multiply when dividing fractions?

A: To invert and multiply when dividing fractions, you need to multiply the first fraction by the reciprocal of the second fraction. This means that you'll multiply the numerator of the first fraction by the denominator of the second fraction, and vice versa.

Q: What is the reciprocal of a fraction?

A: The reciprocal of a fraction is a fraction that has the denominator and numerator swapped. For example, the reciprocal of $\frac{3x}{x^2-x-12}$ is $\frac{x^2-x-12}{3x}$.

Q: How do I cancel out common factors when simplifying a complex expression?

A: To cancel out common factors when simplifying a complex expression, you need to identify any common factors in the numerator and denominator, and then cancel them out. This will simplify the expression and make it easier to work with.

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

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

• Not following the order of operations
• Not factoring the denominators
• Not inverting and multiplying when dividing fractions
• Not canceling out common factors
• Not simplifying the expression step by step

Additional Resources

• Khan Academy: Simplifying Complex Expressions
• Mathway: Simplifying Complex Expressions
• Wolfram Alpha: Simplifying Complex Expressions

Final Thoughts

Simplifying complex expressions is an essential skill that will serve you well in all areas of mathematics. By following the order of operations and simplifying the expression step by step, you'll be able to simplify even the most challenging expressions with ease. Whether you're a student or a professional, mastering the art of simplifying complex expressions is an essential skill that will serve you well in all areas of mathematics.

Conclusion

In this article, we've answered some of the most frequently asked questions about simplifying complex expressions, including dividing fractions. By following the order of operations and simplifying the expression step by step, you'll be able to simplify even the most challenging expressions with ease. Whether you're a student or a professional, mastering the art of simplifying complex expressions is an essential skill that will serve you well in all areas of mathematics.

Frequently Asked Questions

• Q: What is the difference between a fraction and a division? A: A fraction is a way of expressing a part of a whole, while a division is a way of expressing a quotient.
• Q: How do I invert and multiply when dividing fractions? A: To invert and multiply when dividing fractions, you need to multiply the first fraction by the reciprocal of the second fraction.
• Q: What is the reciprocal of a fraction? A: The reciprocal of a fraction is a fraction that has the denominator and numerator swapped.
• Q: How do I cancel out common factors when simplifying a complex expression? A: To cancel out common factors when simplifying a complex expression, you need to identify any common factors in the numerator and denominator, and then cancel them out.

Additional Tips and Tricks

• Make sure to follow the order of operations when simplifying complex expressions.
• Factor the denominators before inverting and multiplying.
• Cancel out common factors to simplify the expression.
• Use the reciprocal of a fraction to invert and multiply.
• Practice, practice, practice! Simplifying complex expressions takes practice, so make sure to practice regularly.

Final Thoughts

Simplifying complex expressions is an essential skill that will serve you well in all areas of mathematics. By following the order of operations and simplifying the expression step by step, you'll be able to simplify even the most challenging expressions with ease. Whether you're a student or a professional, mastering the art of simplifying complex expressions is an essential skill that will serve you well in all areas of mathematics.