Simplify The Rational Expression: ${ \frac{16x^2}{8x} }$Select The Correct Choice Below And, If Necessary, Fill In The Answer Box To Complete Your Choice.A. { \frac{16x^2}{8x} =$}$ { \square$}$ (Simplify Your Answer.)B.

Introduction

Rational expressions are a fundamental concept in algebra, and simplifying them is a crucial skill to master. In this article, we will focus on simplifying the rational expression $\frac{16x^2}{8x}$, and we will explore the step-by-step process involved in simplifying such expressions.

What are Rational Expressions?

A rational expression is a fraction that contains variables and/or constants in the numerator and/or denominator. Rational expressions can be simplified by canceling out common factors between the numerator and denominator.

Simplifying the Rational Expression

To simplify the rational expression $\frac{16x^2}{8x}$, we need to identify the common factors between the numerator and denominator. In this case, both the numerator and denominator have a common factor of $8x$.

Step 1: Factor the Numerator and Denominator

The numerator $16x^2$ can be factored as $8x \cdot 2x$, and the denominator $8x$ can be factored as $8x$.

Step 2: Cancel Out Common Factors

Now that we have factored the numerator and denominator, we can cancel out the common factor of $8x$. This leaves us with $\frac{2x}{1}$.

Step 3: Simplify the Expression

The expression $\frac{2x}{1}$ can be simplified further by canceling out the common factor of $1$. This leaves us with $2x$.

Conclusion

In conclusion, simplifying the rational expression $\frac{16x^2}{8x}$ involves identifying the common factors between the numerator and denominator, factoring the numerator and denominator, canceling out common factors, and simplifying the expression. By following these steps, we can simplify complex rational expressions and arrive at a simplified solution.

Example Problems

Problem 1

Simplify the rational expression $\frac{24x^3}{12x^2}$.

Step 1: Factor the Numerator and Denominator

The numerator $24x^3$ can be factored as $12x^2 \cdot 2x$, and the denominator $12x^2$ can be factored as $12x^2$.

Step 2: Cancel Out Common Factors

Now that we have factored the numerator and denominator, we can cancel out the common factor of $12x^2$. This leaves us with $\frac{2x}{1}$.

Step 3: Simplify the Expression

The expression $\frac{2x}{1}$ can be simplified further by canceling out the common factor of $1$. This leaves us with $2x$.

Solution

The simplified solution is $2x$.

Problem 2

Simplify the rational expression $\frac{36x^4}{18x^3}$.

Step 1: Factor the Numerator and Denominator

The numerator $36x^4$ can be factored as $18x^3 \cdot 2x$, and the denominator $18x^3$ can be factored as $18x^3$.

Step 2: Cancel Out Common Factors

Now that we have factored the numerator and denominator, we can cancel out the common factor of $18x^3$. This leaves us with $\frac{2x}{1}$.

Step 3: Simplify the Expression

The expression $\frac{2x}{1}$ can be simplified further by canceling out the common factor of $1$. This leaves us with $2x$.

Solution

The simplified solution is $2x$.

Tips and Tricks

Tip 1: Identify Common Factors

When simplifying rational expressions, it is essential to identify common factors between the numerator and denominator. This can be done by factoring the numerator and denominator.

Tip 2: Cancel Out Common Factors

Once you have identified common factors, you can cancel them out to simplify the expression.

Tip 3: Simplify the Expression

After canceling out common factors, you can simplify the expression further by canceling out any remaining common factors.

Conclusion

In conclusion, simplifying rational expressions involves identifying common factors, factoring the numerator and denominator, canceling out common factors, and simplifying the expression. By following these steps and using the tips and tricks outlined above, you can simplify complex rational expressions and arrive at a simplified solution.

Final Answer

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Introduction

In our previous article, we explored the step-by-step process of simplifying rational expressions. In this article, we will answer some frequently asked questions about simplifying rational expressions.

Q&A

Q: What is a rational expression?

A: A rational expression is a fraction that contains variables and/or constants in the numerator and/or denominator.

Q: How do I simplify a rational expression?

A: To simplify a rational expression, you need to identify the common factors between the numerator and denominator, factor the numerator and denominator, cancel out common factors, and simplify the expression.

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

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

• Not identifying common factors between the numerator and denominator
• Not factoring the numerator and denominator
• Not canceling out common factors
• Not simplifying the expression

Q: How do I identify common factors between the numerator and denominator?

A: To identify common factors between the numerator and denominator, you need to factor the numerator and denominator. Common factors are factors that appear in both the numerator and denominator.

Q: Can I simplify a rational expression with a variable in the denominator?

A: Yes, you can simplify a rational expression with a variable in the denominator. However, you need to be careful when canceling out common factors, as the variable may not be a common factor.

Q: How do I simplify a rational expression with a negative exponent?

A: To simplify a rational expression with a negative exponent, you need to rewrite the expression with a positive exponent. For example, $\frac{1}{x^{-2}}$ can be rewritten as $x^2$.

Q: Can I simplify a rational expression with a fraction in the denominator?

A: Yes, you can simplify a rational expression with a fraction in the denominator. However, you need to be careful when canceling out common factors, as the fraction may not be a common factor.

Q: How do I simplify a rational expression with multiple variables?

A: To simplify a rational expression with multiple variables, you need to identify the common factors between the numerator and denominator, factor the numerator and denominator, cancel out common factors, and simplify the expression.

Q: Can I simplify a rational expression with a coefficient in the numerator or denominator?

A: Yes, you can simplify a rational expression with a coefficient in the numerator or denominator. However, you need to be careful when canceling out common factors, as the coefficient may not be a common factor.

Example Problems

Problem 1

Simplify the rational expression $\frac{24x^3}{12x^2}$.

Solution

The simplified solution is $2x$.

Problem 2

Simplify the rational expression $\frac{36x^4}{18x^3}$.

Solution

The simplified solution is $2x$.

Tips and Tricks

Tip 1: Identify Common Factors

When simplifying rational expressions, it is essential to identify common factors between the numerator and denominator. This can be done by factoring the numerator and denominator.

Tip 2: Cancel Out Common Factors

Once you have identified common factors, you can cancel them out to simplify the expression.

Tip 3: Simplify the Expression

After canceling out common factors, you can simplify the expression further by canceling out any remaining common factors.

Conclusion

In conclusion, simplifying rational expressions involves identifying common factors, factoring the numerator and denominator, canceling out common factors, and simplifying the expression. By following these steps and using the tips and tricks outlined above, you can simplify complex rational expressions and arrive at a simplified solution.

Final Answer

The final answer is: $\boxed{2x}$