Simplify The Expression:$ \frac{8}{x^2+6x} \cdot 2x }$Options A. { \frac{8 {x+6}$}$B. { \frac{16}{x+6}$}$C. { \frac{8}{x^2+3}$}$D. { \frac{4}{x 3+6x 2}$}$

Introduction

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is an essential skill for students and professionals alike. In this article, we will focus on simplifying a specific expression, $\frac{8}{x^2+6x} \cdot 2x$, and explore the different options available.

Understanding the Expression

The given expression is a product of two terms: $\frac{8}{x^2+6x}$ and $2x$. To simplify this expression, we need to apply the rules of algebra, specifically the distributive property and the properties of exponents.

Step 1: Apply the Distributive Property

The distributive property states that for any real numbers $a$, $b$, and $c$, $a(b+c) = ab + ac$. We can apply this property to the given expression by multiplying the numerator of the first term, $8$, by the second term, $2x$.

\frac{8}{x^2+6x} \cdot 2x = \frac{8 \cdot 2x}{x^2+6x}

Step 2: Simplify the Numerator

Now, we can simplify the numerator by multiplying $8$ and $2x$.

\frac{8 \cdot 2x}{x^2+6x} = \frac{16x}{x^2+6x}

Step 3: Factor the Denominator

The denominator of the expression is a quadratic expression, $x^2+6x$. We can factor this expression by taking out the greatest common factor, $x$.

\frac{16x}{x^2+6x} = \frac{16x}{x(x+6)}

Step 4: Cancel Common Factors

Now, we can cancel out the common factor, $x$, from the numerator and the denominator.

\frac{16x}{x(x+6)} = \frac{16}{x+6}

Conclusion

In conclusion, the simplified expression is $\frac{16}{x+6}$. This is the correct answer among the options provided.

Comparison with Options

Let's compare our simplified expression with the options provided:

• Option A: $\frac{8}{x+6}$ - This is incorrect because the numerator is not $16$.
• Option B: $\frac{16}{x+6}$ - This is correct because it matches our simplified expression.
• Option C: $\frac{8}{x^2+3}$ - This is incorrect because the denominator is not $x+6$.
• Option D: $\frac{4}{x^3+6x^2}$ - This is incorrect because the denominator is not $x+6$.

Final Answer

The final answer is $\boxed{\frac{16}{x+6}}$.

Additional Tips and Tricks

• When simplifying algebraic expressions, always look for common factors to cancel out.
• Use the distributive property to expand expressions and simplify them.
• Factor quadratic expressions by taking out the greatest common factor.
• Cancel out common factors to simplify expressions.

Common Mistakes to Avoid

• Not canceling out common factors when simplifying expressions.
• Not using the distributive property to expand expressions.
• Not factoring quadratic expressions.
• Not simplifying expressions by canceling out common factors.

Conclusion

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Introduction

In our previous article, we explored the steps to simplify the expression $\frac{8}{x^2+6x} \cdot 2x$. In this article, we will provide a Q&A guide to help you understand the concepts and techniques involved in simplifying algebraic expressions.

Q: What is the first step in simplifying an algebraic expression?

A: The first step in simplifying an algebraic expression is to look for common factors to cancel out. This involves identifying any common factors in the numerator and denominator and canceling them out.

Q: How do I apply the distributive property to simplify an expression?

A: To apply the distributive property, you need to multiply the numerator of the first term by the second term. This involves multiplying each term in the numerator by the second term.

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

A: Factoring involves breaking down an expression into its simplest form by identifying any common factors. Simplifying an expression involves reducing it to its simplest form by canceling out any common factors.

Q: How do I factor a quadratic expression?

A: To factor a quadratic expression, you need to identify any common factors and take them out. This involves breaking down the expression into two binomials.

Q: What is the importance of canceling out common factors?

A: Canceling out common factors is essential in simplifying algebraic expressions. It helps to reduce the expression to its simplest form and makes it easier to work with.

Q: Can I simplify an expression by canceling out common factors if there are no common factors?

A: No, you cannot simplify an expression by canceling out common factors if there are no common factors. In this case, the expression is already in its simplest form.

Q: How do I know if an expression is in its simplest form?

A: An expression is in its simplest form if there are no common factors that can be canceled out. This means that the numerator and denominator have no common factors.

Q: Can I simplify an expression by using the distributive property if the expression is already in its simplest form?

A: No, you cannot simplify an expression by using the distributive property if the expression is already in its simplest form. In this case, the expression is already reduced to its simplest form.

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

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

• Not canceling out common factors
• Not using the distributive property to expand expressions
• Not factoring quadratic expressions
• Not simplifying expressions by canceling out common factors

Q: How can I practice simplifying algebraic expressions?

A: You can practice simplifying algebraic expressions by working through examples and exercises. You can also use online resources and practice tests to help you improve your skills.

Conclusion

Simplifying algebraic expressions is an essential skill for students and professionals alike. By following the steps outlined in this article and practicing regularly, you can improve your skills and become more confident in your ability to simplify expressions.

Additional Resources

• Online resources: Khan Academy, Mathway, and Wolfram Alpha
• Practice tests: Algebra.com and Math Open Reference
• Textbooks: Algebra and Trigonometry by Michael Sullivan and Algebra: Structure and Method by Marvin L. Bittinger

Final Tips

• Always look for common factors to cancel out
• Use the distributive property to expand expressions
• Factor quadratic expressions
• Simplify expressions by canceling out common factors
• Practice regularly to improve your skills