Simplify The Expression: A Step-by-Step Guide

Hey guys! Let's break down this math problem together. We're going to simplify the expression: ${ \left( \frac{x+5}{x^2-81} + \frac{x+7}{x^2-18x+81} \right) : \left( \frac{x+3}{x-9} \right)^2 + \frac{7+x}{9+x} }$. Don't worry, we'll take it one step at a time. By the end, you'll not only know the answer but also understand the process. Let's dive in!

1. Factoring the Denominators

Okay, so the first thing we need to do is factor those denominators. Factoring is like reverse multiplication, where we break down an expression into its constituent parts. This will help us identify common factors and simplify the expression more easily. It's like taking a complex puzzle and breaking it into smaller, manageable pieces.

• First denominator: x^2 - 81. Notice that this is a difference of squares. Remember the formula: a^2 - b^2 = (a - b) (a + b). So, x^2 - 81 factors into (x - 9)(x + 9).
• Second denominator: x^2 - 18x + 81. This is a perfect square trinomial. It can be factored as (x - 9)^2. Think of it as (x - 9)(x - 9). Spotting these patterns is key to quick simplification.

So, after factoring, our expression looks like this:

${ \left( \frac{x+5}{(x-9)(x+9)} + \frac{x+7}{(x-9)^2} \right) : \left( \frac{x+3}{x-9} \right)^2 + \frac{7+x}{9+x} }$

2. Finding a Common Denominator

Now that we've factored the denominators, we need to find a common denominator for the fractions inside the parentheses. This will allow us to add the fractions together. It's like making sure you have the same units before you add things up – you can't add apples and oranges directly, right? Similarly, we need a common ground for these fractions.

Looking at the denominators ( x - 9)(x + 9) and (x - 9)^2, the least common denominator (LCD) will be (x - 9)^2 (x + 9). This LCD includes all the factors from both denominators, ensuring we can combine the fractions seamlessly.

• To get the first fraction to have the LCD, we need to multiply both the numerator and denominator by (x - 9): ${ \frac{x+5}{(x-9)(x+9)} \cdot \frac{x-9}{x-9} = \frac{(x+5)(x-9)}{(x-9)^2(x+9)} }$
• To get the second fraction to have the LCD, we need to multiply both the numerator and denominator by (x + 9): ${ \frac{x+7}{(x-9)^2} \cdot \frac{x+9}{x+9} = \frac{(x+7)(x+9)}{(x-9)^2(x+9)} }$

Our expression now looks like:

${ \left( \frac{(x+5)(x-9)}{(x-9)^2(x+9)} + \frac{(x+7)(x+9)}{(x-9)^2(x+9)} \right) : \left( \frac{x+3}{x-9} \right)^2 + \frac{7+x}{9+x} }$

3. Adding the Fractions

With a common denominator in place, we can now add the fractions inside the parentheses. This involves combining the numerators while keeping the denominator the same. Think of it like adding slices of a pizza – if they're all cut into the same size (common denominator), you just add the number of slices (numerators).

So, we add the numerators: ${ (x+5)(x-9) + (x+7)(x+9) }$

Let's expand these products: ${ (x^2 - 9x + 5x - 45) + (x^2 + 9x + 7x + 63) }$

Combine like terms: ${ x^2 - 4x - 45 + x^2 + 16x + 63 = 2x^2 + 12x + 18 }$

Now, we can rewrite the expression inside the parentheses:

${ \frac{2x^2 + 12x + 18}{(x-9)^2(x+9)} }$

Notice that we can factor out a 2 from the numerator:

${ \frac{2(x^2 + 6x + 9)}{(x-9)^2(x+9)} }$

And the quadratic expression in the numerator is another perfect square trinomial:

${ x^2 + 6x + 9 = (x+3)^2 }$

So, the expression inside the parentheses simplifies to:

${ \frac{2(x+3)^2}{(x-9)^2(x+9)} }$

Our full expression now looks like this:

{ \frac{2(x+3)^2}{(x-9)^2(x+9)} : \left( rac{x+3}{x-9} ight)^2 + \frac{7+x}{9+x} }

4. Dividing the Fractions

Next up, we're dividing fractions. Remember, dividing by a fraction is the same as multiplying by its reciprocal (flipping the fraction). It's like asking how many halves are in a whole – you're essentially multiplying by 2.

We're dividing by ${ \left( \frac{x+3}{x-9} \right)^2 }$, which is the same as ${ \frac{(x+3)^2}{(x-9)^2} }$. The reciprocal of this is ${ \frac{(x-9)^2}{(x+3)^2} }$.

So, we rewrite the division as multiplication:

${ \frac{2(x+3)^2}{(x-9)^2(x+9)} \cdot \frac{(x-9)^2}{(x+3)^2} + \frac{7+x}{9+x} }$

Now, we can cancel out common factors. Notice that ${ (x+3)^2 }$ and ${ (x-9)^2 }$ appear in both the numerator and the denominator, so they cancel out:

${ \frac{2}{(x+9)} + \frac{7+x}{9+x} }$

5. Adding the Remaining Fractions

We're almost there! Now we just need to add the remaining fractions. Lucky for us, they already have a common denominator, which is (x + 9). So, we can directly add the numerators.

${ \frac{2}{x+9} + \frac{7+x}{9+x} = \frac{2 + (7+x)}{x+9} }$

Combine the terms in the numerator:

${ \frac{9+x}{x+9} }$

6. Final Simplification

And finally, we have ${ \frac{9+x}{x+9} }$. Notice that the numerator and denominator are the same! Anything divided by itself is 1 (as long as the expression is defined, which means x cannot be -9).

So, the simplified expression is:

${ 1 }$

Therefore, the correct answer is B. 1.

Conclusion

Guys, we did it! We took a complex expression and simplified it step-by-step. Remember, the key is to break the problem down into smaller parts: factor, find common denominators, add/subtract, multiply/divide, and simplify. Keep practicing, and you'll become a pro at simplifying expressions in no time! If you have any questions, feel free to ask. Happy simplifying!