Simplify The Given Expression And Write It As A Single Fraction. Factor The Numerator And Denominator If Possible.${ \frac{5}{x+6} + 4 = \square }$(Factor Completely. Simplify Your Answer. Use Positive Exponents Only.)

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Introduction

In this article, we will delve into the world of algebra and simplify a given expression by writing it as a single fraction. We will factor the numerator and denominator if possible and use positive exponents only. This will involve a series of steps, including isolating the fraction, combining like terms, and simplifying the resulting expression.

Step 1: Isolate the Fraction

The given expression is 5x+6+4\frac{5}{x+6} + 4. To simplify this expression, we need to isolate the fraction. We can do this by subtracting 4 from both sides of the equation.

5x+6+4=5x+6+4(x+6)x+6\frac{5}{x+6} + 4 = \frac{5}{x+6} + \frac{4(x+6)}{x+6}

Step 2: Combine Like Terms

Now that we have isolated the fraction, we can combine like terms. We can rewrite the expression as a single fraction by finding a common denominator.

5x+6+4(x+6)x+6=5+4(x+6)x+6\frac{5}{x+6} + \frac{4(x+6)}{x+6} = \frac{5 + 4(x+6)}{x+6}

Step 3: Simplify the Numerator

The next step is to simplify the numerator. We can do this by distributing the 4 to the terms inside the parentheses.

5+4(x+6)x+6=5+4x+24x+6\frac{5 + 4(x+6)}{x+6} = \frac{5 + 4x + 24}{x+6}

Step 4: Combine Like Terms in the Numerator

Now that we have simplified the numerator, we can combine like terms.

5+4x+24x+6=29+4xx+6\frac{5 + 4x + 24}{x+6} = \frac{29 + 4x}{x+6}

Step 5: Factor the Numerator and Denominator

The final step is to factor the numerator and denominator if possible. In this case, we can factor the numerator as a difference of squares.

29+4xx+6=(4x+1)2βˆ’1x+6\frac{29 + 4x}{x+6} = \frac{(4x+1)^2 - 1}{x+6}

Step 6: Simplify the Expression

Now that we have factored the numerator and denominator, we can simplify the expression. We can rewrite the expression as a single fraction by canceling out any common factors.

(4x+1)2βˆ’1x+6=(4x+1)2βˆ’12x+6\frac{(4x+1)^2 - 1}{x+6} = \frac{(4x+1)^2 - 1^2}{x+6}

Step 7: Factor the Difference of Squares

The final step is to factor the difference of squares.

(4x+1)2βˆ’12x+6=(4x+1+1)(4x+1βˆ’1)x+6\frac{(4x+1)^2 - 1^2}{x+6} = \frac{(4x+1+1)(4x+1-1)}{x+6}

Step 8: Simplify the Expression

Now that we have factored the difference of squares, we can simplify the expression. We can rewrite the expression as a single fraction by canceling out any common factors.

(4x+1+1)(4x+1βˆ’1)x+6=(4x+2)(4x)x+6\frac{(4x+1+1)(4x+1-1)}{x+6} = \frac{(4x+2)(4x)}{x+6}

Step 9: Factor the Numerator

The final step is to factor the numerator.

(4x+2)(4x)x+6=2(2x+1)(2x)x+6\frac{(4x+2)(4x)}{x+6} = \frac{2(2x+1)(2x)}{x+6}

Step 10: Simplify the Expression

Now that we have factored the numerator, we can simplify the expression. We can rewrite the expression as a single fraction by canceling out any common factors.

2(2x+1)(2x)x+6=2(2x+1)(2x)(x+3)(x+2)\frac{2(2x+1)(2x)}{x+6} = \frac{2(2x+1)(2x)}{(x+3)(x+2)}

Step 11: Cancel Out Common Factors

The final step is to cancel out any common factors.

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

Conclusion

In this article, we simplified the given expression by writing it as a single fraction. We factored the numerator and denominator if possible and used positive exponents only. The final simplified expression is 2(2x+1)(2x)(x+3)(x+2)\frac{2(2x+1)(2x)}{(x+3)(x+2)}.

Final Answer

The final answer is 2(2x+1)(2x)(x+3)(x+2)\boxed{\frac{2(2x+1)(2x)}{(x+3)(x+2)}}.

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Introduction

In our previous article, we simplified the given expression by writing it as a single fraction. We factored the numerator and denominator if possible and used positive exponents only. In this article, we will answer some frequently asked questions related to simplifying the given expression.

Q&A

Q: What is the first step in simplifying the given expression?

A: The first step in simplifying the given expression is to isolate the fraction. We can do this by subtracting 4 from both sides of the equation.

Q: How do I combine like terms in the numerator?

A: To combine like terms in the numerator, we need to distribute the 4 to the terms inside the parentheses. This will allow us to simplify the numerator and rewrite the expression as a single fraction.

Q: Can I factor the numerator and denominator if possible?

A: Yes, we can factor the numerator and denominator if possible. In this case, we can factor the numerator as a difference of squares.

Q: How do I simplify the expression after factoring the numerator and denominator?

A: After factoring the numerator and denominator, we can simplify the expression by canceling out any common factors. This will allow us to rewrite the expression as a single fraction.

Q: What is the final simplified expression?

A: The final simplified expression is 2(2x+1)(2x)(x+3)(x+2)\frac{2(2x+1)(2x)}{(x+3)(x+2)}.

Q: Can I cancel out common factors in the numerator and denominator?

A: Yes, we can cancel out common factors in the numerator and denominator. This will allow us to simplify the expression and rewrite it as a single fraction.

Q: What is the importance of using positive exponents only?

A: Using positive exponents only is important because it allows us to simplify the expression and rewrite it as a single fraction. It also makes it easier to work with the expression and perform calculations.

Q: Can I use this method to simplify other expressions?

A: Yes, we can use this method to simplify other expressions. The steps involved in simplifying the given expression are general and can be applied to other expressions as well.

Common Mistakes

Mistake 1: Not isolating the fraction

  • Not isolating the fraction can make it difficult to simplify the expression and rewrite it as a single fraction.
  • To avoid this mistake, make sure to isolate the fraction by subtracting 4 from both sides of the equation.

Mistake 2: Not combining like terms

  • Not combining like terms can make it difficult to simplify the numerator and rewrite the expression as a single fraction.
  • To avoid this mistake, make sure to combine like terms by distributing the 4 to the terms inside the parentheses.

Mistake 3: Not factoring the numerator and denominator

  • Not factoring the numerator and denominator can make it difficult to simplify the expression and rewrite it as a single fraction.
  • To avoid this mistake, make sure to factor the numerator and denominator if possible.

Conclusion

In this article, we answered some frequently asked questions related to simplifying the given expression. We also discussed some common mistakes that can be made when simplifying the expression. By following the steps involved in simplifying the given expression and avoiding common mistakes, we can simplify the expression and rewrite it as a single fraction.

Final Answer

The final answer is 2(2x+1)(2x)(x+3)(x+2)\boxed{\frac{2(2x+1)(2x)}{(x+3)(x+2)}}.