Simplify The Expression: ${ \frac{x-2}{x^2+4x-5} \div \frac{x 2-4}{x 2+5x} }$
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Introduction
When it comes to simplifying complex expressions, one of the most challenging tasks is dividing fractions. In this article, we will explore the step-by-step process of simplifying the given expression: ${ \frac{x-2}{x^2+4x-5} \div \frac{x2-4}{x2+5x} }$
Understanding the Concept of Dividing Fractions
Dividing fractions involves inverting the second fraction and then multiplying it by the first fraction. This concept is based on the rule that division is the same as multiplying by the reciprocal of the divisor. In other words, when we divide one fraction by another, we can simply multiply the first fraction by the reciprocal of the second fraction.
Step 1: Invert the Second Fraction
To simplify the given expression, we need to start by inverting the second fraction. This means that we will change the sign of the numerator and denominator of the second fraction. The inverted second fraction becomes: ${ \frac{x2+5x}{x2-4} }$
Step 2: Multiply the First Fraction by the Inverted Second Fraction
Now that we have inverted the second fraction, we can multiply it by the first fraction. This will give us a new fraction that is the result of the division. The multiplication process involves multiplying the numerators and denominators separately. The resulting fraction becomes: ${ \frac{(x-2)(x2+5x)}{(x2+4x-5)(x^2-4)} }$
Step 3: Simplify the Numerator and Denominator
To simplify the resulting fraction, we need to factor the numerator and denominator. Factoring involves breaking down the expressions into their prime factors. The numerator can be factored as: ${ (x-2)(x^2+5x) = x3+3x2-10x }$
The denominator can be factored as: ${ (x2+4x-5)(x2-4) = (x+5)(x-1)(x-2)(x+2) }$
Step 4: Cancel Out Common Factors
Now that we have factored the numerator and denominator, we can cancel out any common factors. In this case, we can cancel out the common factor of (x-2) from the numerator and denominator. The resulting fraction becomes: ${ \frac{x3+3x2-10x}{(x+5)(x-1)(x+2)} }$
Conclusion
Simplifying the given expression involves inverting the second fraction, multiplying it by the first fraction, and then simplifying the resulting fraction. By following these steps, we can simplify the expression and arrive at the final answer. The simplified expression is: ${ \frac{x3+3x2-10x}{(x+5)(x-1)(x+2)} }$
Final Answer
The final answer is: ${ \frac{x3+3x2-10x}{(x+5)(x-1)(x+2)} }$
Frequently Asked Questions
Q: What is the concept of dividing fractions?
A: Dividing fractions involves inverting the second fraction and then multiplying it by the first fraction.
Q: How do I simplify the numerator and denominator?
A: To simplify the numerator and denominator, you need to factor them into their prime factors.
Q: Can I cancel out common factors?
A: Yes, you can cancel out common factors from the numerator and denominator.
Q: What is the final answer?
A: The final answer is: ${ \frac{x3+3x2-10x}{(x+5)(x-1)(x+2)} }$
Additional Resources
For more information on simplifying expressions and dividing fractions, you can refer to the following resources:
- Khan Academy: Dividing Fractions
- Mathway: Simplifying Expressions
- Wolfram Alpha: Dividing Fractions
Conclusion
Simplifying the given expression involves inverting the second fraction, multiplying it by the first fraction, and then simplifying the resulting fraction. By following these steps, we can simplify the expression and arrive at the final answer. The simplified expression is: ${
\frac{x3+3x2-10x}{(x+5)(x-1)(x+2)}
}$
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Q&A: Simplifying Expressions and Dividing Fractions
Q: What is the concept of dividing fractions?
A: Dividing fractions involves inverting the second fraction and then multiplying it by the first fraction. This concept is based on the rule that division is the same as multiplying by the reciprocal of the divisor.
Q: How do I simplify the numerator and denominator?
A: To simplify the numerator and denominator, you need to factor them into their prime factors. This involves breaking down the expressions into their simplest form.
Q: Can I cancel out common factors?
A: Yes, you can cancel out common factors from the numerator and denominator. This is an important step in simplifying the expression.
Q: What is the final answer?
A: The final answer is: ${ \frac{x3+3x2-10x}{(x+5)(x-1)(x+2)} }$
Q: How do I know if I have simplified the expression correctly?
A: To check if you have simplified the expression correctly, you can plug in a value for x and see if the expression evaluates to the correct value.
Q: Can I use a calculator to simplify the expression?
A: Yes, you can use a calculator to simplify the expression. However, it's always a good idea to check your work by hand to make sure you understand the process.
Q: What if I have a fraction with a variable in the denominator?
A: If you have a fraction with a variable in the denominator, you can simplify it by factoring the denominator and canceling out any common factors.
Q: Can I simplify a fraction with a negative exponent?
A: Yes, you can simplify a fraction with a negative exponent by rewriting it as a positive exponent and then simplifying.
Q: How do I simplify a fraction with a variable in the numerator and denominator?
A: To simplify a fraction with a variable in the numerator and denominator, you need to factor the numerator and denominator and then cancel out any common factors.
Q: Can I use a graphing calculator to simplify the expression?
A: Yes, you can use a graphing calculator to simplify the expression. However, it's always a good idea to check your work by hand to make sure you understand the process.
Additional Resources
For more information on simplifying expressions and dividing fractions, you can refer to the following resources:
- Khan Academy: Dividing Fractions
- Mathway: Simplifying Expressions
- Wolfram Alpha: Dividing Fractions
Conclusion
Simplifying expressions and dividing fractions can be a challenging task, but with practice and patience, you can master these skills. Remember to always follow the steps outlined in this article and to check your work by hand to make sure you understand the process.
Final Tips
- Always start by simplifying the numerator and denominator separately.
- Use factoring to simplify the numerator and denominator.
- Cancel out common factors to simplify the expression.
- Check your work by hand to make sure you understand the process.
- Use a calculator or graphing calculator to check your work and get a visual representation of the expression.
By following these tips and practicing regularly, you can become proficient in simplifying expressions and dividing fractions.