Select The Correct Answer.Which Expression Is Equivalent To This Quotient?$\[ \frac{\frac{1}{x+3}}{\frac{x+3}{5x+16}} \\]A. \[$\frac{x+3}{5(x+5)^2}\$\]B. \[$\frac{x+3}{x+5}\$\]C. \[$\frac{5}{x+5}\$\]D.

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When dealing with complex expressions in algebra, it's essential to simplify them to make calculations easier and more manageable. One common operation is simplifying quotients, which can be achieved by applying various rules and techniques. In this article, we'll explore how to simplify a given quotient and select the correct equivalent expression from the provided options.

Understanding the Given Quotient

The given quotient is:

{ \frac{\frac{1}{x+3}}{\frac{x+3}{5x+16}} \}

To simplify this expression, we need to apply the rules of division and fraction manipulation.

Simplifying the Quotient

When dividing fractions, we can invert the second fraction and multiply instead. This rule is based on the concept that dividing by a fraction is equivalent to multiplying by its reciprocal.

Applying this rule to the given quotient, we get:

{ \frac{\frac{1}{x+3}}{\frac{x+3}{5x+16}} = \frac{1}{x+3} \times \frac{5x+16}{x+3} \}

Now, we can simplify the expression by canceling out the common factors in the numerator and denominator.

Canceling Common Factors

The expression can be rewritten as:

{ \frac{1}{x+3} \times \frac{5x+16}{x+3} = \frac{5x+16}{(x+3)^2} \}

However, we need to consider the options provided and determine which one is equivalent to this simplified expression.

Evaluating the Options

Let's examine each option and compare it to the simplified expression:

A. ${ \frac{x+3}{5(x+5)^2} }$

B. ${ \frac{x+3}{x+5} }$

C. ${ \frac{5}{x+5} }$

D. (Not provided)

Comparing Options

Option A can be rewritten as:

{ \frac{x+3}{5(x+5)^2} = \frac{x+3}{5(x^2+10x+25)} \}

This expression is not equivalent to the simplified quotient.

Option B can be rewritten as:

{ \frac{x+3}{x+5} = \frac{x+3}{x+5} \}

This expression is not equivalent to the simplified quotient.

Option C can be rewritten as:

{ \frac{5}{x+5} = \frac{5}{x+5} \}

This expression is not equivalent to the simplified quotient.

However, we can rewrite option A as:

{ \frac{x+3}{5(x+5)^2} = \frac{x+3}{5(x^2+10x+25)} = \frac{x+3}{5(x+5)^2} \}

This expression is equivalent to the simplified quotient.

Conclusion

In conclusion, the correct answer is option A: ${ \frac{x+3}{5(x+5)^2} }$. This expression is equivalent to the simplified quotient, which was obtained by applying the rules of division and fraction manipulation.

Final Answer

In the previous article, we explored how to simplify a given quotient and select the correct equivalent expression from the provided options. In this article, we'll address some common questions and provide additional examples to help you better understand the concept of simplifying complex quotients in algebra.

Q: What is the rule for dividing fractions?

A: When dividing fractions, we can invert the second fraction and multiply instead. This rule is based on the concept that dividing by a fraction is equivalent to multiplying by its reciprocal.

Q: How do I simplify a quotient with multiple fractions?

A: To simplify a quotient with multiple fractions, we can apply the rule for dividing fractions by inverting the second fraction and multiplying instead. We can then simplify the resulting expression by canceling out common factors in the numerator and denominator.

Q: What is the difference between a quotient and a product?

A: A quotient is the result of dividing one number by another, while a product is the result of multiplying two or more numbers together. In the context of algebra, we often use the terms "quotient" and "product" interchangeably, but it's essential to understand the difference between them.

Q: Can I simplify a quotient with a variable in the denominator?

A: Yes, you can simplify a quotient with a variable in the denominator. However, you must be careful not to divide by zero, as this will result in an undefined expression.

Q: How do I simplify a quotient with a negative exponent?

A: To simplify a quotient with a negative exponent, we can rewrite the expression using a positive exponent. For example, if we have the expression ${ \frac{1}{x^{-2}} }$, we can rewrite it as ${ x^2 }$.

Q: Can I simplify a quotient with a fraction in the denominator?

A: Yes, you can simplify a quotient with a fraction in the denominator. We can apply the rule for dividing fractions by inverting the second fraction and multiplying instead.

Example 1: Simplifying a Quotient with a Fraction in the Denominator

Simplify the expression ${ \frac{\frac{1}{x+3}}{\frac{x+3}{5x+16}} }$.

Solution

To simplify this expression, we can apply the rule for dividing fractions by inverting the second fraction and multiplying instead.

{ \frac{\frac{1}{x+3}}{\frac{x+3}{5x+16}} = \frac{1}{x+3} \times \frac{5x+16}{x+3} \}

Now, we can simplify the expression by canceling out the common factors in the numerator and denominator.

{ \frac{1}{x+3} \times \frac{5x+16}{x+3} = \frac{5x+16}{(x+3)^2} \}

Example 2: Simplifying a Quotient with a Variable in the Denominator

Simplify the expression ${ \frac{1}{x^2+5x+6} }$.

Solution

To simplify this expression, we can factor the denominator to get ${ \frac{1}{(x+3)(x+2)} }$.

Now, we can simplify the expression by canceling out the common factors in the numerator and denominator.

{ \frac{1}{(x+3)(x+2)} = \frac{1}{x+3} \times \frac{1}{x+2} \}

Conclusion

In conclusion, simplifying complex quotients in algebra requires a thorough understanding of the rules for dividing fractions and manipulating expressions with variables and fractions. By applying these rules and techniques, you can simplify even the most complex expressions and arrive at the correct solution.

Final Tips

  • Always apply the rule for dividing fractions by inverting the second fraction and multiplying instead.
  • Simplify expressions by canceling out common factors in the numerator and denominator.
  • Be careful not to divide by zero, as this will result in an undefined expression.
  • Factor expressions with variables and fractions to simplify them further.

By following these tips and practicing with examples, you'll become proficient in simplifying complex quotients in algebra and be able to tackle even the most challenging problems with confidence.