\[$\frac{x^2-16}{x^2+8x+16}\$\] Is Equivalent To:A. \[$\frac{x+4}{x-4}\$\]B. \[$\frac{x-4}{x-4}\$\]C. \[$\frac{x+4}{x+4}\$\]D. \[$\frac{x-4}{x+4}\$\]

Feb 26, 2025 by ADMIN 150 views

Simplifying Rational Expressions: A Step-by-Step Guide

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Introduction

Rational expressions are a fundamental concept in algebra, and simplifying them is a crucial skill for any math enthusiast. In this article, we will delve into the world of rational expressions and explore the process of simplifying them. We will use the given expression ${$ \frac{x^2-16}{x2+8x+16}$}$ as a case study and demonstrate how to simplify it step by step.

Understanding 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. The goal of simplifying a rational expression is to reduce it to its simplest form, which is a fraction with no common factors between the numerator and denominator.

Simplifying the Given Expression

To simplify the given expression ${$ \frac{x^2-16}{x2+8x+16}$}$, we need to factor both the numerator and denominator. The numerator can be factored as ${$ (x+4)(x-4)$}$, and the denominator can be factored as ${$ (x+4)(x+4)$}$.

import sympy as sp

# Define the variables
x = sp.symbols('x')

# Define the numerator and denominator
numerator = x**2 - 16
denominator = x**2 + 8*x + 16

# Factor the numerator and denominator
factored_numerator = sp.factor(numerator)
factored_denominator = sp.factor(denominator)

print(factored_numerator)
print(factored_denominator)

Canceling Out Common Factors

Now that we have factored the numerator and denominator, we can cancel out the common factors. The numerator has a factor of ${$ (x+4)$}$, and the denominator has a factor of ${$ (x+4)$}$. We can cancel out these common factors by dividing both the numerator and denominator by ${$ (x+4)$}$.

# Cancel out the common factors
simplified_expression = factored_numerator / factored_denominator

print(simplified_expression)

Simplifying the Expression Further

After canceling out the common factors, we are left with the simplified expression ${$ \frac{x-4}{x+4}$}$. However, we can simplify this expression further by factoring the numerator and denominator.

# Factor the numerator and denominator
factored_numerator = sp.factor(simplified_expression.as_numer_denom()[0])
factored_denominator = sp.factor(simplified_expression.as_numer_denom()[1])

print(factored_numerator)
print(factored_denominator)

Conclusion

In conclusion, simplifying rational expressions is a crucial skill for any math enthusiast. By following the steps outlined in this article, we can simplify even the most complex rational expressions. The given expression ${$ \frac{x^2-16}{x2+8x+16}$}$ was simplified step by step, and the final simplified expression was ${$ \frac{x-4}{x+4}$}$.

Final Answer

The final answer is ${$ \frac{x-4}{x+4}$}$.

Discussion

The given expression ${$ \frac{x^2-16}{x2+8x+16}$}$ is equivalent to ${$ \frac{x-4}{x+4}$}$. This can be verified by simplifying the expression step by step, as demonstrated in this article.

Common Mistakes

When simplifying rational expressions, it is easy to make mistakes. Some common mistakes include:

Not factoring the numerator and denominator
Not canceling out common factors
Not simplifying the expression further

Tips and Tricks

When simplifying rational expressions, here are some tips and tricks to keep in mind:

Always factor the numerator and denominator
Cancel out common factors whenever possible
Simplify the expression further by factoring the numerator and denominator

Real-World Applications

Simplifying rational expressions has many real-world applications. Some examples include:

Simplifying complex fractions in finance and economics
Simplifying rational expressions in physics and engineering
Simplifying rational expressions in computer science and programming

Conclusion

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Introduction

In our previous article, we explored the process of simplifying rational expressions. We used the given expression ${$ \frac{x^2-16}{x2+8x+16}$}$ as a case study and demonstrated how to simplify it step by step. 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: Why is it important to simplify rational expressions?

A: Simplifying rational expressions is important because it helps to reduce the complexity of the expression and makes it easier to work with. It also helps to eliminate any common factors that may be present in the numerator and denominator.

Q: How do I simplify a rational expression?

A: To simplify a rational expression, you need to factor the numerator and denominator, cancel out any common factors, and simplify the expression further if possible.

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, not canceling out common factors, and not simplifying the expression further.

Q: How do I factor a rational expression?

A: To factor a rational expression, you need to identify any common factors in the numerator and denominator and factor them out.

Q: What is the difference between a rational expression and a rational number?

A: A rational number is a number that can be expressed as a fraction, whereas a rational expression is a fraction that contains variables and/or constants in the numerator and/or 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 to avoid introducing any extraneous solutions.

Q: How do I know if a rational expression is already in its simplest form?

A: To determine if a rational expression is already in its simplest form, you need to check if there are any common factors between the numerator and denominator. If there are no common factors, then the expression is already in its simplest form.

Q: Can I simplify a rational expression with a negative exponent?

A: Yes, you can simplify a rational expression with a negative exponent. However, you need to be careful when canceling out common factors to avoid introducing any extraneous solutions.

Q: How do I simplify a rational expression with a fraction in the numerator or denominator?

A: To simplify a rational expression with a fraction in the numerator or denominator, you need to multiply the numerator and denominator by the reciprocal of the fraction.

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

A: Yes, you can simplify a rational expression with a variable in the numerator and a constant in the denominator. However, you need to be careful when canceling out common factors to avoid introducing any extraneous solutions.

Conclusion

In conclusion, simplifying rational expressions is a crucial skill for any math enthusiast. By following the steps outlined in this article and answering the frequently asked questions, we can simplify even the most complex rational expressions. Remember to always factor the numerator and denominator, cancel out common factors, and simplify the expression further if possible.

Final Tips

Always factor the numerator and denominator
Cancel out common factors whenever possible
Simplify the expression further by factoring the numerator and denominator
Be careful when canceling out common factors to avoid introducing any extraneous solutions
Check if the expression is already in its simplest form before simplifying it further

Real-World Applications

Simplifying rational expressions has many real-world applications. Some examples include:

Simplifying complex fractions in finance and economics
Simplifying rational expressions in physics and engineering
Simplifying rational expressions in computer science and programming