Simplify The Expression:$\[ \frac{15}{y^2 + 2y - 8} \div \frac{5y}{y-2} \\]

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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 15y2+2y−8÷5yy−2\frac{15}{y^2 + 2y - 8} \div \frac{5y}{y-2} using a step-by-step approach.

Understanding the Expression

Before we dive into the simplification process, let's take a closer look at the expression we're working with. The expression is a division of two fractions, which can be written as:

15y2+2y−8÷5yy−2\frac{15}{y^2 + 2y - 8} \div \frac{5y}{y-2}

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 Denominator

The first step in simplifying the expression is to factor the denominator of the first fraction. The denominator is a quadratic expression that can be factored as follows:

y2+2y−8=(y+4)(y−2)y^2 + 2y - 8 = (y + 4)(y - 2)

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

15(y+4)(y−2)÷5yy−2\frac{15}{(y + 4)(y - 2)} \div \frac{5y}{y-2}

Step 2: Cancel Common Factors

The next step is to cancel any common factors between the numerator and denominator of the two fractions. In this case, we can cancel the (y−2)(y - 2) factor between the two fractions:

15(y+4)(y−2)÷5yy−2=15(y+4)÷5y1\frac{15}{(y + 4)(y - 2)} \div \frac{5y}{y-2} = \frac{15}{(y + 4)} \div \frac{5y}{1}

Step 3: Invert and Multiply

Now that we've cancelled any common factors, we can invert the second fraction and multiply:

15(y+4)÷5y1=15(y+4)×15y\frac{15}{(y + 4)} \div \frac{5y}{1} = \frac{15}{(y + 4)} \times \frac{1}{5y}

Step 4: Multiply the Numerators and Denominators

The final step is to multiply the numerators and denominators of the two fractions:

15(y+4)×15y=15×1(y+4)×5y\frac{15}{(y + 4)} \times \frac{1}{5y} = \frac{15 \times 1}{(y + 4) \times 5y}

Step 5: Simplify the Expression

Now that we've multiplied the numerators and denominators, we can simplify the expression by cancelling any common factors:

15×1(y+4)×5y=3(y+4)×y\frac{15 \times 1}{(y + 4) \times 5y} = \frac{3}{(y + 4) \times y}

And that's it! We've successfully simplified the expression 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 15y2+2y−8÷5yy−2\frac{15}{y^2 + 2y - 8} \div \frac{5y}{y-2} using a step-by-step approach. By following the order of operations and cancelling any common factors, we were able to simplify the expression to 3(y+4)×y\frac{3}{(y + 4) \times y}.

Frequently Asked Questions

  • Q: What is the order of operations? A: The order of operations is a set of rules that dictate 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 cancel any common factors between the numerator and denominator of the two fractions.
  • Q: What is the difference between a fraction and a decimal? A: A fraction is a mathematical expression that represents a part of a whole, while a decimal is a numerical value that represents a part of a whole.

Additional Resources

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

Final Thoughts

Simplifying complex expressions is an essential skill for anyone who wants to succeed in mathematics. By following the order of operations and cancelling any common factors, you'll be able to simplify even the most challenging expressions with ease. Remember to practice regularly and seek help when you need it. With time and practice, you'll become a master of simplifying complex expressions.

Introduction

In our previous article, we took a closer look at how to simplify the expression 15y2+2y−8÷5yy−2\frac{15}{y^2 + 2y - 8} \div \frac{5y}{y-2} 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.

Q&A: Simplifying Complex Expressions

Q: What is the order of operations?

A: The order of operations is a set of rules that dictate 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 cancel any common factors between the numerator and denominator of the two fractions.

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

A: A fraction is a mathematical expression that represents a part of a whole, while a decimal is a numerical value that represents a part of a whole.

Q: Can I simplify a complex expression by just canceling out the common factors?

A: No, you cannot simplify a complex expression by just canceling out the common factors. You must follow the order of operations and perform the operations in the correct order.

Q: What if I have a complex expression with multiple fractions?

A: If you have a complex expression with multiple fractions, you can simplify it by following the order of operations and canceling any common factors between the numerator and denominator of the two fractions.

Q: Can I use a calculator to simplify a complex expression?

A: Yes, you can use a calculator to simplify a complex expression. However, it's always a good idea to double-check your work by following the order of operations and canceling any common factors.

Q: What if I get stuck on a complex expression?

A: If you get stuck on a complex expression, don't be afraid to ask for help. You can ask a teacher, tutor, or classmate for assistance. You can also use online resources, such as Khan Academy or Mathway, to get help with simplifying complex expressions.

Tips and Tricks for Simplifying Complex Expressions

  • Always follow the order of operations.
  • Cancel any common factors between the numerator and denominator of the two fractions.
  • Use a calculator to check your work.
  • Don't be afraid to ask for help if you get stuck.
  • Practice regularly to improve your skills.

Common Mistakes to Avoid

  • Not following the order of operations.
  • Canceling out common factors without checking the numerator and denominator.
  • Not using a calculator to check your work.
  • Being afraid to ask for help.

Conclusion

Simplifying complex expressions can be a challenging task, but with practice and patience, you'll become a master of simplifying complex expressions. Remember to follow the order of operations, cancel any common factors, and use a calculator to check your work. Don't be afraid to ask for help if you get stuck, and practice regularly to improve your skills.

Frequently Asked Questions

  • Q: What is the order of operations? A: The order of operations is a set of rules that dictate 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 cancel any common factors between the numerator and denominator of the two fractions.
  • Q: What is the difference between a fraction and a decimal? A: A fraction is a mathematical expression that represents a part of a whole, while a decimal is a numerical value that represents a part of a whole.

Additional Resources

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

Final Thoughts

Simplifying complex expressions is an essential skill for anyone who wants to succeed in mathematics. By following the order of operations and canceling any common factors, you'll be able to simplify even the most challenging expressions with ease. Remember to practice regularly and seek help when you need it. With time and practice, you'll become a master of simplifying complex expressions.