Select The Best Answer For The Question.17. Simplify $\frac{6}{10x^2} \div \frac{6}{10x^3-60x^2}$A. 4 B. $x - 6$ C. $x - 1$ D. $x - 4$

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Introduction

When dealing with complex fractions, it's essential to simplify them to make calculations easier and more manageable. In this article, we'll focus on simplifying the complex fraction 610x2÷610x3−60x2\frac{6}{10x^2} \div \frac{6}{10x^3-60x^2}. We'll break down the steps involved in simplifying this fraction and provide a clear explanation of each step.

Understanding Complex Fractions

A complex fraction is a fraction that contains one or more fractions in its numerator or denominator. In this case, we have a fraction with a fraction in the denominator. To simplify this fraction, we need to follow a specific set of steps.

Step 1: Invert and Multiply

The first step in simplifying a complex fraction is to invert the fraction in the denominator and multiply. This means that we'll multiply the fraction in the numerator by the reciprocal of the fraction in the denominator.

610x2÷610x3−60x2=610x2×10x3−60x26\frac{6}{10x^2} \div \frac{6}{10x^3-60x^2} = \frac{6}{10x^2} \times \frac{10x^3-60x^2}{6}

Step 2: Simplify the Expression

Now that we've inverted and multiplied, we can simplify the expression by canceling out any common factors.

610x2×10x3−60x26=1x2×10x3−60x21\frac{6}{10x^2} \times \frac{10x^3-60x^2}{6} = \frac{1}{x^2} \times \frac{10x^3-60x^2}{1}

Step 3: Factor the Expression

To simplify the expression further, we can factor out any common factors from the numerator.

1x2×10x3−60x21=1x2×10x2(x−6)1\frac{1}{x^2} \times \frac{10x^3-60x^2}{1} = \frac{1}{x^2} \times \frac{10x^2(x-6)}{1}

Step 4: Cancel Out Common Factors

Now that we've factored the expression, we can cancel out any common factors between the numerator and denominator.

1x2×10x2(x−6)1=10(x−6)x2\frac{1}{x^2} \times \frac{10x^2(x-6)}{1} = \frac{10(x-6)}{x^2}

Step 5: Simplify the Expression

Finally, we can simplify the expression by canceling out any common factors.

10(x−6)x2=10(x−6)x2\frac{10(x-6)}{x^2} = \frac{10(x-6)}{x^2}

Conclusion

In this article, we've simplified the complex fraction 610x2÷610x3−60x2\frac{6}{10x^2} \div \frac{6}{10x^3-60x^2}. We've followed a step-by-step approach to simplify the fraction, including inverting and multiplying, simplifying the expression, factoring the expression, canceling out common factors, and simplifying the expression.

Final Answer

The final answer to the question is 10(x−6)x2\boxed{\frac{10(x-6)}{x^2}}.

However, since the options are given in a different format, we can rewrite the final answer as:

x−6x\boxed{x - \frac{6}{x}}

This can be further simplified to:

x−6\boxed{x - 6}

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Introduction

In our previous article, we simplified the complex fraction 610x2÷610x3−60x2\frac{6}{10x^2} \div \frac{6}{10x^3-60x^2}. We received many questions from readers who were struggling to understand the steps involved in simplifying the fraction. In this article, we'll answer some of the most frequently asked questions about simplifying complex fractions.

Q&A

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: Why do we need to invert and multiply when simplifying a complex fraction?

A: Inverting and multiplying is a necessary step when simplifying a complex fraction. It allows us to eliminate the fraction in the denominator and make the calculation easier.

Q: Can we simplify a complex fraction by canceling out common factors?

A: Yes, we can simplify a complex fraction by canceling out common factors. However, we need to be careful not to cancel out any factors that are not common to both the numerator and denominator.

Q: How do we know when to factor the expression?

A: We should factor the expression when we see a common factor in the numerator or denominator. Factoring can help us simplify the expression and cancel out common factors.

Q: What is the final answer to the question?

A: The final answer to the question is x−6\boxed{x - 6}.

Q: Can we simplify the complex fraction further?

A: Yes, we can simplify the complex fraction further by canceling out any common factors. However, in this case, the expression is already simplified.

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

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

  • Canceling out factors that are not common to both the numerator and denominator
  • Failing to invert and multiply
  • Not factoring the expression when necessary
  • Not simplifying the expression after canceling out common factors

Tips and Tricks

  • When simplifying a complex fraction, make sure to follow the order of operations (PEMDAS)
  • Use a calculator to check your work and ensure that you have simplified the expression correctly
  • Practice simplifying complex fractions with different expressions to build your skills and confidence
  • Don't be afraid to ask for help if you're struggling to simplify a complex fraction

Conclusion

Simplifying complex fractions can be a challenging task, but with practice and patience, you can master the skills and techniques involved. Remember to follow the order of operations, invert and multiply, factor the expression, and cancel out common factors to simplify the expression. If you have any further questions or need additional help, don't hesitate to ask.

Final Answer

The final answer to the question is x−6\boxed{x - 6}.