Simplify The Expression:e) $\frac{7 Y^2}{y^2-9} \times \frac{4 Y+12}{14 Y^3}$

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Introduction

Simplifying algebraic expressions is a crucial skill in mathematics, and it's essential to understand the steps involved in simplifying complex expressions. In this article, we will focus on simplifying the expression $\frac{7 y^2}{y^2-9} \times \frac{4 y+12}{14 y^3}$. We will break down the expression into smaller parts, simplify each part, and then combine them to get the final simplified expression.

Understanding the Expression

The given expression is a product of two fractions: $\frac{7 y^2}{y^2-9}$ and $\frac{4 y+12}{14 y^3}$. To simplify this expression, we need to first simplify each fraction individually and then multiply them together.

Simplifying the First Fraction

The first fraction is $\frac{7 y^2}{y^2-9}$. We can simplify this fraction by factoring the denominator. The denominator $y^2-9$ can be factored as $(y+3)(y-3)$. Therefore, the first fraction can be written as $\frac{7 y^2}{(y+3)(y-3)}$.

Simplifying the Second Fraction

The second fraction is $\frac{4 y+12}{14 y^3}$. We can simplify this fraction by factoring the numerator and the denominator. The numerator $4 y+12$ can be factored as $4(y+3)$, and the denominator $14 y^3$ can be factored as $2 \times 7 \times y^3$. Therefore, the second fraction can be written as $\frac{4(y+3)}{2 \times 7 \times y^3}$.

Multiplying the Fractions

Now that we have simplified each fraction, we can multiply them together to get the final expression. When multiplying fractions, we multiply the numerators together and the denominators together. Therefore, the product of the two fractions is:

$$\frac{7 y^2}{(y+3)(y-3)} \times \frac{4(y+3)}{2 \times 7 \times y^3} = \frac{7 y^2 \times 4(y+3)}{(y+3)(y-3) \times 2 \times 7 \times y^3}$$

Canceling Common Factors

Now that we have multiplied the fractions, we can simplify the expression by canceling common factors. The expression has several common factors that can be canceled out. The $(y+3)$ terms in the numerator and the denominator can be canceled out, as well as the $7$ terms in the numerator and the denominator. Therefore, the expression simplifies to:

$$\frac{7 y^2 \times 4(y+3)}{(y+3)(y-3) \times 2 \times 7 \times y^3} = \frac{4 y^2}{(y-3) \times 2 \times y^3}$$

Final Simplification

The expression can be further simplified by canceling out the common factor of $y^2$ in the numerator and the denominator. Therefore, the final simplified expression is:

$$\frac{4 y^2}{(y-3) \times 2 \times y^3} = \frac{2}{(y-3) \times y}$$

Conclusion

In this article, we simplified the expression $\frac{7 y^2}{y^2-9} \times \frac{4 y+12}{14 y^3}$ by breaking it down into smaller parts, simplifying each part, and then combining them to get the final simplified expression. We used factoring, canceling common factors, and simplifying to arrive at the final answer. This process demonstrates the importance of simplifying algebraic expressions and the steps involved in simplifying complex expressions.

Frequently Asked Questions

• Q: What is the final simplified expression? A: The final simplified expression is $\frac{2}{(y-3) \times y}$.
• Q: How do I simplify an algebraic expression? A: To simplify an algebraic expression, you need to break it down into smaller parts, simplify each part, and then combine them to get the final simplified expression.
• Q: What are the steps involved in simplifying an algebraic expression? A: The steps involved in simplifying an algebraic expression include factoring, canceling common factors, and simplifying.

Additional Resources

• For more information on simplifying algebraic expressions, please refer to the following resources:

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

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Introduction

In our previous article, we simplified the expression $\frac{7 y^2}{y^2-9} \times \frac{4 y+12}{14 y^3}$ by breaking it down into smaller parts, simplifying each part, and then combining them to get the final simplified expression. In this article, we will answer some frequently asked questions about simplifying algebraic expressions and provide additional resources for further learning.

Q&A

Q: What is the final simplified expression?

A: The final simplified expression is $\frac{2}{(y-3) \times y}$.

Q: How do I simplify an algebraic expression?

A: To simplify an algebraic expression, you need to break it down into smaller parts, simplify each part, and then combine them to get the final simplified expression.

Q: What are the steps involved in simplifying an algebraic expression?

A: The steps involved in simplifying an algebraic expression include factoring, canceling common factors, and simplifying.

Q: Can I simplify an algebraic expression by canceling out terms?

A: Yes, you can simplify an algebraic expression by canceling out terms. However, you need to make sure that the terms you are canceling out are actually common factors.

Q: How do I know if a term is a common factor?

A: A term is a common factor if it appears in both the numerator and the denominator of the expression.

Q: Can I simplify an algebraic expression by combining like terms?

A: Yes, you can simplify an algebraic expression by combining like terms. However, you need to make sure that the terms you are combining are actually like terms.

Q: How do I know if two terms are like terms?

A: Two terms are like terms if they have the same variable and exponent.

Q: Can I simplify an algebraic expression by using a calculator?

A: Yes, you can simplify an algebraic expression by using a calculator. However, you need to make sure that the calculator is set to the correct mode and that you are entering the expression correctly.

Q: How do I know if an algebraic expression is already simplified?

A: An algebraic expression is already simplified if it cannot be simplified further by factoring, canceling common factors, or combining like terms.

Additional Resources

• For more information on simplifying algebraic expressions, please refer to the following resources:

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

• For practice problems and exercises, please refer to the following resources:

• IXL: Algebraic Expressions
• Math Open Reference: Algebraic Expressions
• Algebra.com: Algebraic Expressions

• For online calculators and tools, please refer to the following resources:

• Wolfram Alpha: Algebraic Expressions
• Mathway: Algebraic Expressions
• Desmos: Algebraic Expressions

Conclusion

In this article, we answered some frequently asked questions about simplifying algebraic expressions and provided additional resources for further learning. We hope that this article has been helpful in understanding the steps involved in simplifying algebraic expressions and in providing additional resources for further learning.

Frequently Asked Questions

• Q: What is the final simplified expression? A: The final simplified expression is $\frac{2}{(y-3) \times y}$.
• Q: How do I simplify an algebraic expression? A: To simplify an algebraic expression, you need to break it down into smaller parts, simplify each part, and then combine them to get the final simplified expression.
• Q: What are the steps involved in simplifying an algebraic expression? A: The steps involved in simplifying an algebraic expression include factoring, canceling common factors, and simplifying.

Additional Tips

• Make sure to read the problem carefully and understand what is being asked.
• Break down the problem into smaller parts and simplify each part.
• Use factoring, canceling common factors, and combining like terms to simplify the expression.
• Check your work by plugging in values or using a calculator.
• Practice, practice, practice! The more you practice, the better you will become at simplifying algebraic expressions.