Assuming No Denominator Equals Zero, Which Expression Is Equivalent To The Given Expression? R − 5 R 2 5 − R 3 R 2 \frac{\frac{r-5}{r^2}}{\frac{5-r}{3r^2}} 3 R 2 5 − R ​ R 2 R − 5 ​ ​ A. − 3 7 -\frac{3}{7} − 7 3 ​ B. T 3 \frac{t}{3} 3 T ​ C. − Π 3 -\frac{\pi}{3} − 3 Π ​ D.

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Introduction

When dealing with complex fractions, it can be challenging to determine which expression is equivalent to the given expression. In this article, we will explore the process of simplifying complex fractions and apply this knowledge to the given expression: r−5r25−r3r2\frac{\frac{r-5}{r^2}}{\frac{5-r}{3r^2}}. We will examine each option and determine which one is equivalent to the given expression.

Understanding Complex Fractions

A complex fraction is a fraction that contains one or more fractions in its numerator or denominator. To simplify a complex fraction, we need to follow a specific set of steps. The first step is to simplify the numerator and denominator separately. This involves combining like terms and simplifying any fractions within the numerator or denominator.

Simplifying the Given Expression

To simplify the given expression, we need to follow the steps outlined above. The given expression is: r−5r25−r3r2\frac{\frac{r-5}{r^2}}{\frac{5-r}{3r^2}}. The first step is to simplify the numerator and denominator separately.

Simplifying the Numerator

The numerator of the given expression is: r−5r2\frac{r-5}{r^2}. This fraction cannot be simplified further, so we will leave it as is.

Simplifying the Denominator

The denominator of the given expression is: 5−r3r2\frac{5-r}{3r^2}. This fraction cannot be simplified further, so we will leave it as is.

Simplifying the Complex Fraction

Now that we have simplified the numerator and denominator separately, we can simplify the complex fraction. To do this, we need to multiply the numerator and denominator by the reciprocal of the denominator.

r−5r25−r3r2=r−5r2⋅3r25−r\frac{\frac{r-5}{r^2}}{\frac{5-r}{3r^2}} = \frac{r-5}{r^2} \cdot \frac{3r^2}{5-r}

Canceling Out Common Factors

Now that we have multiplied the numerator and denominator by the reciprocal of the denominator, we can cancel out any common factors. In this case, we can cancel out the r2r^2 term in the numerator and denominator.

r−5r2⋅3r25−r=3(r−5)5−r\frac{r-5}{r^2} \cdot \frac{3r^2}{5-r} = \frac{3(r-5)}{5-r}

Simplifying the Expression

Now that we have canceled out the common factors, we can simplify the expression further. To do this, we need to multiply the numerator and denominator by the reciprocal of the denominator.

3(r−5)5−r=3(r−5)5−r⋅−(5−r)−(5−r)\frac{3(r-5)}{5-r} = \frac{3(r-5)}{5-r} \cdot \frac{-(5-r)}{-(5-r)}

Canceling Out Common Factors

Now that we have multiplied the numerator and denominator by the reciprocal of the denominator, we can cancel out any common factors. In this case, we can cancel out the (5−r)(5-r) term in the numerator and denominator.

3(r−5)5−r⋅−(5−r)−(5−r)=−3\frac{3(r-5)}{5-r} \cdot \frac{-(5-r)}{-(5-r)} = -3

Conclusion

In conclusion, the expression r−5r25−r3r2\frac{\frac{r-5}{r^2}}{\frac{5-r}{3r^2}} is equivalent to the expression −3-3. This is the correct answer.

Discussion

The given expression is a complex fraction that contains one or more fractions in its numerator and denominator. To simplify this expression, we need to follow a specific set of steps. The first step is to simplify the numerator and denominator separately. This involves combining like terms and simplifying any fractions within the numerator or denominator. Once we have simplified the numerator and denominator separately, we can simplify the complex fraction by multiplying the numerator and denominator by the reciprocal of the denominator. Finally, we can cancel out any common factors and simplify the expression further.

Final Answer

The final answer is: −3\boxed{-3}

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Introduction

In our previous article, we explored the process of simplifying complex fractions and applied this knowledge to the given expression: r−5r25−r3r2\frac{\frac{r-5}{r^2}}{\frac{5-r}{3r^2}}. We determined that the expression is equivalent to the expression −3-3. In this article, we will provide a Q&A guide to help you better understand the process of simplifying complex fractions.

Q&A Guide

Q: What is a complex fraction?

A: A complex fraction is a fraction that contains one or more fractions in its numerator or denominator.

Q: How do I simplify a complex fraction?

A: To simplify a complex fraction, you need to follow a specific set of steps. The first step is to simplify the numerator and denominator separately. This involves combining like terms and simplifying any fractions within the numerator or denominator. Once you have simplified the numerator and denominator separately, you can simplify the complex fraction by multiplying the numerator and denominator by the reciprocal of the denominator.

Q: What is the reciprocal of a fraction?

A: The reciprocal of a fraction is obtained by swapping the numerator and denominator. For example, the reciprocal of ab\frac{a}{b} is ba\frac{b}{a}.

Q: How do I multiply a fraction by its reciprocal?

A: To multiply a fraction by its reciprocal, you can simply swap the numerator and denominator. For example, ab⋅ba=1\frac{a}{b} \cdot \frac{b}{a} = 1.

Q: What is the difference between a complex fraction and a nested fraction?

A: A complex fraction is a fraction that contains one or more fractions in its numerator or denominator. A nested fraction is a fraction that contains one or more fractions within its numerator or denominator.

Q: How do I simplify a nested fraction?

A: To simplify a nested fraction, you need to follow a specific set of steps. The first step is to simplify the innermost fraction. Once you have simplified the innermost fraction, you can simplify the outer fraction.

Q: What are some common mistakes to avoid when simplifying complex fractions?

A: Some common mistakes to avoid when simplifying complex fractions include:

  • Not simplifying the numerator and denominator separately
  • Not multiplying the numerator and denominator by the reciprocal of the denominator
  • Not canceling out common factors
  • Not simplifying the expression further

Conclusion

In conclusion, simplifying complex fractions can be a challenging task, but with the right steps and techniques, you can simplify even the most complex expressions. By following the Q&A guide provided in this article, you can better understand the process of simplifying complex fractions and avoid common mistakes.

Final Tips

  • Always simplify the numerator and denominator separately
  • Always multiply the numerator and denominator by the reciprocal of the denominator
  • Always cancel out common factors
  • Always simplify the expression further

Final Answer

The final answer is: −3\boxed{-3}