Simplify: 27 X 3 − 8 27 X 2 + 18 X + 12 \frac{27x^3-8}{27x^2+18x+12} 27 X 2 + 18 X + 12 27 X 3 − 8 ​

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Introduction

Simplifying rational expressions is a crucial skill in algebra, and it's essential to understand the process to solve various mathematical problems. In this article, we will focus on simplifying the given rational expression $\frac{27x^3-8}{27x^2+18x+12}$. We will break down the steps involved in simplifying this expression and provide a clear explanation of each step.

Step 1: Factor the Numerator and Denominator

To simplify the rational expression, we need to factor both the numerator and the denominator. The numerator is $27x^3-8$, and the denominator is $27x^2+18x+12$. We can start by factoring the numerator.

Factoring the Numerator

The numerator $27x^3-8$ can be factored as follows:

$$27x^3-8 = (3x-1)(9x^2+3x+8)$$

We can see that the numerator has been factored into two parts: $(3x-1)$ and $(9x^2+3x+8)$.

Factoring the Denominator

The denominator $27x^2+18x+12$ can be factored as follows:

$$27x^2+18x+12 = 3(9x^2+6x+4) = 3(3x+2)(3x+2) = 3(3x+2)^2$$

We can see that the denominator has been factored into three parts: $3$, $(3x+2)$, and $(3x+2)$.

Step 2: Cancel Common Factors

Now that we have factored both the numerator and the denominator, we can cancel common factors. We can see that both the numerator and the denominator have a common factor of $(3x-1)$.

Canceling Common Factors

We can cancel the common factor $(3x-1)$ from both the numerator and the denominator as follows:

$$\frac{(3x-1)(9x^2+3x+8)}{3(3x+2)^2} = \frac{9x^2+3x+8}{3(3x+2)^2}$$

We can see that the common factor $(3x-1)$ has been canceled from both the numerator and the denominator.

Step 3: Simplify the Expression

Now that we have canceled the common factor, we can simplify the expression further. We can see that the denominator $3(3x+2)^2$ can be simplified as follows:

Simplifying the Denominator

The denominator $3(3x+2)^2$ can be simplified as follows:

$$3(3x+2)^2 = 3(9x^2+12x+4) = 27x^2+36x+12$$

We can see that the denominator has been simplified.

Step 4: Write the Final Answer

Now that we have simplified the expression, we can write the final answer. We can see that the simplified expression is:

$$\frac{9x^2+3x+8}{27x^2+36x+12}$$

We can see that the final answer is a simplified rational expression.

Conclusion

In this article, we have simplified the rational expression $\frac{27x^3-8}{27x^2+18x+12}$. We have broken down the steps involved in simplifying this expression and provided a clear explanation of each step. We have factored the numerator and denominator, canceled common factors, and simplified the expression. We have written the final answer as a simplified rational expression. We hope that this article has provided a clear understanding of how to simplify rational expressions.

Frequently Asked Questions

• Q: What is the process of simplifying rational expressions? A: The process of simplifying rational expressions involves factoring the numerator and denominator, canceling common factors, and simplifying the expression.
• Q: How do I factor the numerator and denominator? A: To factor the numerator and denominator, you need to find the greatest common factor (GCF) of the terms and factor it out.
• Q: What is the difference between factoring and canceling common factors? A: Factoring involves breaking down an expression into its simplest form, while canceling common factors involves removing common factors from both the numerator and the denominator.

Final Answer

The final answer is $\boxed{\frac{9x^2+3x+8}{27x^2+36x+12}}$.

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Introduction

In our previous article, we simplified the rational expression $\frac{27x^3-8}{27x^2+18x+12}$. We broke down the steps involved in simplifying this expression and provided a clear explanation of each step. In this article, we will answer some frequently asked questions related to simplifying rational expressions.

Q&A

Q: What is the process of simplifying rational expressions?

A: The process of simplifying rational expressions involves factoring the numerator and denominator, canceling common factors, and simplifying the expression.

Q: How do I factor the numerator and denominator?

A: To factor the numerator and denominator, you need to find the greatest common factor (GCF) of the terms and factor it out. You can use the following steps to factor the numerator and denominator:

• Find the GCF of the terms in the numerator and denominator.
• Factor out the GCF from each term in the numerator and denominator.
• Simplify the expression by canceling out any common factors.

Q: What is the difference between factoring and canceling common factors?

A: Factoring involves breaking down an expression into its simplest form, while canceling common factors involves removing common factors from both the numerator and the denominator.

Q: Can I simplify a rational expression if it has no common factors?

A: Yes, you can simplify a rational expression even if it has no common factors. You can simplify the expression by factoring the numerator and denominator and then canceling out any common factors.

Q: How do I know if a rational expression can be simplified?

A: You can check if a rational expression can be simplified by factoring the numerator and denominator and then canceling out any common factors. If you can simplify the expression, then it can be simplified.

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

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

• Not factoring the numerator and denominator correctly.
• Not canceling out common factors correctly.
• Not simplifying the expression correctly.

Q: Can I use a calculator to simplify rational expressions?

A: Yes, you can use a calculator to simplify rational expressions. However, it's always a good idea to check your work by hand to make sure that the expression is simplified correctly.

Q: How do I check my work when simplifying rational expressions?

A: You can check your work by simplifying the expression by hand and then comparing it to the simplified expression obtained using a calculator. If the two expressions are the same, then you have simplified the expression correctly.

Conclusion

In this article, we have answered some frequently asked questions related to simplifying rational expressions. We have provided a clear explanation of the process of simplifying rational expressions and have highlighted some common mistakes to avoid. We hope that this article has provided a clear understanding of how to simplify rational expressions.

Final Answer

The final answer is $\boxed{\frac{9x^2+3x+8}{27x^2+36x+12}}$.

Additional Resources

• For more information on simplifying rational expressions, please see our previous article on the topic.
• For more practice problems on simplifying rational expressions, please see our practice problem section.
• For more resources on algebra, please see our resources section.

Practice Problems

• Simplify the rational expression $\frac{2x^2+5x+3}{x^2+2x+1}$.
• Simplify the rational expression $\frac{3x^2+2x+1}{2x^2+3x+2}$.
• Simplify the rational expression $\frac{x^2+4x+3}{x^2+2x+1}$.

Solutions

• The simplified expression is $\frac{2x+3}{x+1}$.
• The simplified expression is $\frac{3x+1}{2x+2}$.
• The simplified expression is $\frac{x+3}{x+1}$.

Final Answer

The final answer is $\boxed{\frac{9x^2+3x+8}{27x^2+36x+12}}$.