Simplify Completely:${ \frac{2+\frac{20}{x}}{1-\frac{100}{x^2}} }$Enter The Numerator And Denominator Separately In The Boxes Below. If The Denominator Is 1, Enter The Number 1. Do Not Leave Either Box Blank. Make Sure That The Coefficient

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Introduction

Rational expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill to master. In this article, we will delve into the world of rational expressions and provide a step-by-step guide on how to simplify the given expression: 2+20x1−100x2\frac{2+\frac{20}{x}}{1-\frac{100}{x^2}}. We will break down the process into manageable parts, making it easy to understand and follow along.

Understanding Rational Expressions

Before we dive into the simplification process, let's take a moment to understand what rational expressions are. 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 in the numerator and denominator.

Simplifying the Expression

To simplify the given expression, we need to focus on the numerator and denominator separately. The numerator is 2+20x2+\frac{20}{x}, and the denominator is 1−100x21-\frac{100}{x^2}.

Simplifying the Numerator

Let's start by simplifying the numerator. We can rewrite the numerator as follows:

2+20x=2xx+20x=2x+20x2+\frac{20}{x} = \frac{2x}{x} + \frac{20}{x} = \frac{2x + 20}{x}

We can simplify the numerator further by factoring out the common factor of 2:

2x+20x=2(x+10)x\frac{2x + 20}{x} = \frac{2(x + 10)}{x}

Simplifying the Denominator

Now, let's simplify the denominator. We can rewrite the denominator as follows:

1−100x2=x2x2−100x2=x2−100x21-\frac{100}{x^2} = \frac{x^2}{x^2} - \frac{100}{x^2} = \frac{x^2 - 100}{x^2}

We can simplify the denominator further by factoring out the common factor of x2x^2:

x2−100x2=(x+10)(x−10)x2\frac{x^2 - 100}{x^2} = \frac{(x + 10)(x - 10)}{x^2}

Combining the Simplified Numerator and Denominator

Now that we have simplified the numerator and denominator, we can combine them to get the simplified expression:

2(x+10)x÷(x+10)(x−10)x2\frac{2(x + 10)}{x} \div \frac{(x + 10)(x - 10)}{x^2}

To divide fractions, we need to invert the second fraction and multiply:

2(x+10)x×x2(x+10)(x−10)\frac{2(x + 10)}{x} \times \frac{x^2}{(x + 10)(x - 10)}

We can cancel out the common factor of (x+10)(x + 10) in the numerator and denominator:

2x2x(x−10)\frac{2x^2}{x(x - 10)}

Final Simplification

We can simplify the expression further by canceling out the common factor of xx in the numerator and denominator:

2xx−10\frac{2x}{x - 10}

Conclusion

Simplifying rational expressions can be a challenging task, but with practice and patience, it becomes easier. By breaking down the process into manageable parts and focusing on the numerator and denominator separately, we can simplify even the most complex rational expressions. In this article, we simplified the given expression: 2+20x1−100x2\frac{2+\frac{20}{x}}{1-\frac{100}{x^2}} and arrived at the final simplified expression: 2xx−10\frac{2x}{x - 10}. We hope this article has provided valuable insights and practical tips on how to simplify rational expressions.

Frequently Asked Questions

  • 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, focus on the numerator and denominator separately and cancel out common factors.
  • Q: What is the final simplified expression for the given problem? A: The final simplified expression is 2xx−10\frac{2x}{x - 10}.

Additional Resources

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

Final Thoughts

Simplifying rational expressions is an essential skill in mathematics, and with practice and patience, it becomes easier. By following the step-by-step guide provided in this article, you can simplify even the most complex rational expressions. Remember to focus on the numerator and denominator separately, cancel out common factors, and arrive at the final simplified expression. Happy simplifying!

Introduction

Rational expressions are a fundamental concept in mathematics, and simplifying them can be a challenging task. In our previous article, we provided a step-by-step guide on how to simplify the expression: 2+20x1−100x2\frac{2+\frac{20}{x}}{1-\frac{100}{x^2}}. In this article, we will address some of the most frequently asked questions about rational expressions and provide answers to help you better understand this concept.

Q&A: Rational Expressions

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, focus on the numerator and denominator separately and cancel out common factors.

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 the ratio of two integers, 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 dividing by zero.

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

A: A rational expression is simplified when there are no common factors between the numerator and denominator that can be canceled out.

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 applying the rules of exponents to avoid errors.

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, focus on the numerator and denominator separately and cancel out common factors.

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

A: Yes, you can simplify a rational expression with a variable in the numerator and denominator. However, you need to be careful when canceling out common factors to avoid dividing by zero.

Q: How do I know if a rational expression is undefined?

A: A rational expression is undefined when the denominator is equal to zero.

Q: Can I simplify a rational expression with a complex fraction?

A: Yes, you can simplify a rational expression with a complex fraction. However, you need to be careful when applying the rules of fractions to avoid errors.

Additional Resources

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

Final Thoughts

Simplifying rational expressions can be a challenging task, but with practice and patience, it becomes easier. By following the step-by-step guide provided in this article and addressing some of the most frequently asked questions, you can better understand this concept and simplify even the most complex rational expressions. Remember to focus on the numerator and denominator separately, cancel out common factors, and arrive at the final simplified expression. Happy simplifying!

Rational Expression Simplification Rules

  • Simplify the numerator and denominator separately.
  • Cancel out common factors between the numerator and denominator.
  • Be careful when canceling out common factors to avoid dividing by zero.
  • Simplify the expression by applying the rules of fractions and exponents.
  • Check if the denominator is equal to zero before simplifying the expression.

Rational Expression Simplification Examples

  • 2+20x1−100x2=2xx−10\frac{2+\frac{20}{x}}{1-\frac{100}{x^2}} = \frac{2x}{x - 10}
  • x2+4x+4x2−4=(x+2)2(x+2)(x−2)\frac{x^2 + 4x + 4}{x^2 - 4} = \frac{(x + 2)^2}{(x + 2)(x - 2)}
  • 3x2−12x+9x2−9=(3x−3)2(x−3)(x+3)\frac{3x^2 - 12x + 9}{x^2 - 9} = \frac{(3x - 3)^2}{(x - 3)(x + 3)}

Rational Expression Simplification Practice Problems

  • Simplify the expression: 2+30x1−90x2\frac{2+\frac{30}{x}}{1-\frac{90}{x^2}}
  • Simplify the expression: x2+6x+9x2−9\frac{x^2 + 6x + 9}{x^2 - 9}
  • Simplify the expression: 4x2−16x+16x2−16\frac{4x^2 - 16x + 16}{x^2 - 16}