Simplify The Expression:${ \frac{2x+10}{(x-3)^2} \div \frac{(x+5) 3}{x 2-9} }$

Introduction

When it comes to simplifying complex mathematical expressions, dividing fractions can be a daunting task. However, with a clear understanding of the rules and a step-by-step approach, you can simplify even the most intricate expressions. In this article, we will focus on simplifying the expression $\frac{2x+10}{(x-3)^2} \div \frac{(x+5)^3}{x^2-9}$.

Understanding the Basics of Dividing Fractions

Before we dive into the solution, let's review the basics of dividing fractions. When dividing fractions, we need to invert the second fraction (i.e., flip the numerator and denominator) and then multiply the fractions. This is based on the rule that division is the same as multiplying by the reciprocal of the divisor.

Step 1: Invert the Second Fraction

To simplify the given expression, we need to invert the second fraction. This means we will flip the numerator and denominator of the second fraction.

$\frac{(x+5)^3}{x^2-9} \rightarrow \frac{x^2-9}{(x+5)^3}$

Step 2: Multiply the Fractions

Now that we have inverted the second fraction, we can multiply the fractions together. To do this, we need to multiply the numerators and denominators separately.

$\frac{2x+10}{(x-3)^2} \cdot \frac{x^2-9}{(x+5)^3} = \frac{(2x+10)(x^2-9)}{(x-3)^2(x+5)^3}$

Step 3: Simplify the Expression

Now that we have multiplied the fractions, we can simplify the expression by factoring the numerator and denominator.

$\frac{(2x+10)(x^2-9)}{(x-3)^2(x+5)^3} = \frac{(2x+10)(x-3)(x+3)}{(x-3)^2(x+5)^3}$

Step 4: Cancel Out Common Factors

Now that we have factored the numerator and denominator, we can cancel out common factors. In this case, we can cancel out the $(x-3)$ term in the numerator and denominator.

$\frac{(2x+10)(x-3)(x+3)}{(x-3)^2(x+5)^3} = \frac{(2x+10)(x+3)}{(x-3)(x+5)^3}$

Step 5: Final Simplification

Now that we have cancelled out the common factors, we can simplify the expression further by combining like terms in the numerator.

$\frac{(2x+10)(x+3)}{(x-3)(x+5)^3} = \frac{2x^2+16x+30}{(x-3)(x+5)^3}$

Conclusion

In conclusion, simplifying the expression $\frac{2x+10}{(x-3)^2} \div \frac{(x+5)^3}{x^2-9}$ requires a step-by-step approach. By inverting the second fraction, multiplying the fractions, factoring the numerator and denominator, and cancelling out common factors, we can simplify the expression to $\frac{2x^2+16x+30}{(x-3)(x+5)^3}$. This example demonstrates the importance of following the rules of dividing fractions and simplifying complex mathematical expressions.

Frequently Asked Questions

• Q: What is the rule for dividing fractions? A: The rule for dividing fractions is to invert the second fraction and then multiply the fractions.
• Q: How do I simplify a complex mathematical expression? A: To simplify a complex mathematical expression, you need to follow the order of operations (PEMDAS) and simplify each step of the way.
• Q: What is the difference between inverting a fraction and multiplying by the reciprocal? A: Inverting a fraction means flipping the numerator and denominator, while multiplying by the reciprocal means multiplying by the fraction's reciprocal.

Final Thoughts

Simplifying complex mathematical expressions requires patience, practice, and a clear understanding of the rules. By following the steps outlined in this article, you can simplify even the most intricate expressions and become more confident in your mathematical abilities. Remember to always follow the order of operations and simplify each step of the way to ensure accuracy and precision.

Introduction

In our previous article, we explored the step-by-step process of simplifying the expression $\frac{2x+10}{(x-3)^2} \div \frac{(x+5)^3}{x^2-9}$. However, we understand that sometimes, the best way to learn is through asking questions and receiving answers. In this article, we will address some of the most frequently asked questions about simplifying complex mathematical expressions, with a focus on dividing fractions.

Q&A: Simplifying Complex Mathematical Expressions

Q: What is the rule for dividing fractions?

A: The rule for dividing fractions is to invert the second fraction and then multiply the fractions. This means that you need to flip the numerator and denominator of the second fraction and then multiply the fractions together.

Q: How do I simplify a complex mathematical expression?

A: To simplify a complex mathematical expression, you need to follow the order of operations (PEMDAS) and simplify each step of the way. This means that you need to:

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

Q: What is the difference between inverting a fraction and multiplying by the reciprocal?

A: Inverting a fraction means flipping the numerator and denominator, while multiplying by the reciprocal means multiplying by the fraction's reciprocal. For example, if you have the fraction $\frac{1}{2}$, inverting it would give you $\frac{2}{1}$, while multiplying by the reciprocal would give you $2 \cdot \frac{1}{2} = 1$.

Q: Can I simplify a fraction by canceling out common factors?

A: Yes, you can simplify a fraction by canceling out common factors. This means that if you have a fraction with a numerator and denominator that have common factors, you can cancel out those factors to simplify the fraction.

Q: How do I know when to simplify a fraction?

A: You should simplify a fraction whenever possible. This means that if you have a fraction with common factors in the numerator and denominator, you should cancel them out to simplify the fraction.

Q: Can I simplify a complex mathematical expression by using a calculator?

A: While a calculator can be a useful tool for simplifying complex mathematical expressions, it is not always the best option. This is because a calculator may not always be able to simplify an expression in the same way that a human can. Additionally, using a calculator can sometimes lead to errors or oversimplification of the expression.

Q: How do I know if a fraction is in its simplest form?

A: A fraction is in its simplest form when the numerator and denominator have no common factors other than 1. This means that if you have a fraction with a numerator and denominator that have no common factors, it is in its simplest form.

Conclusion

Simplifying complex mathematical expressions can be a challenging task, but with practice and patience, you can become more confident in your abilities. By following the rules outlined in this article and using the Q&A guide, you can simplify even the most intricate expressions and become more proficient in your mathematical skills.

Frequently Asked Questions

• Q: What is the rule for dividing fractions? A: The rule for dividing fractions is to invert the second fraction and then multiply the fractions.
• Q: How do I simplify a complex mathematical expression? A: To simplify a complex mathematical expression, you need to follow the order of operations (PEMDAS) and simplify each step of the way.
• Q: What is the difference between inverting a fraction and multiplying by the reciprocal? A: Inverting a fraction means flipping the numerator and denominator, while multiplying by the reciprocal means multiplying by the fraction's reciprocal.

Final Thoughts

Simplifying complex mathematical expressions requires patience, practice, and a clear understanding of the rules. By following the steps outlined in this article and using the Q&A guide, you can simplify even the most intricate expressions and become more confident in your mathematical abilities. Remember to always follow the order of operations and simplify each step of the way to ensure accuracy and precision.