Question

1. \\(\frac{3}{p+4} + \frac{2p+32}{p^2 - 16}\\)\
2. \\(\frac{n^2 + 7n - 18}{1} - \frac{1}{n^2 - 5n + 6} \cdot \frac{1}{n - 3}\\)\
3. \\(\frac{2}{y - 7} + \frac{1}{y - 4}\\)\
4. \\(\frac{r}{r^2 - 9} - \frac{1}{r^2 + 8r + 15}\\)\

directions: simplify each complex fraction.\

1. \\(\frac{4 - \frac{1}{k}}{\frac{8}{k} - \frac{2}{k^2}}\\)\
2. \\(3 + \frac{\frac{v}{2}}{\frac{v}{4} - \frac{9}{v}}\\)\
3. \\(\frac{2a}{9b} - \frac{b}{2a} \div \frac{a^3 + 1}{3b^3 + 2b}\\)\
4. \\(x^2 + \frac{1}{x} \div \frac{(x + 1)^2}{4x}\\)\
5. write an expression to represent the area of the rectangle below in simplest form. one side: \\(\frac{2x^2 - 32x}{x + 4}\\), the other: \\(\frac{1 - \frac{4}{x^2}}{4(x - 4)^2}\\)

Explanation:

Let's solve problem 10: $\frac{3}{p + 4}+\frac{2p + 32}{p^{2}-16}$

Step 1: Factor the denominator

Notice that $p^{2}-16$ is a difference of squares, so $p^{2}-16=(p + 4)(p - 4)$
The expression becomes $\frac{3}{p + 4}+\frac{2p + 32}{(p + 4)(p - 4)}$

Step 2: Find a common denominator

The common denominator for the two fractions is $(p + 4)(p - 4)$
Rewrite the first fraction with the common denominator: $\frac{3(p - 4)}{(p + 4)(p - 4)}+\frac{2p + 32}{(p + 4)(p - 4)}$

Step 3: Expand the numerator of the first fraction

$3(p - 4)=3p-12$
So the expression is $\frac{3p - 12}{(p + 4)(p - 4)}+\frac{2p + 32}{(p + 4)(p - 4)}$

Step 4: Add the numerators

$(3p - 12)+(2p + 32)=3p-12 + 2p+32=5p + 20$
The fraction becomes $\frac{5p + 20}{(p + 4)(p - 4)}$

Step 5: Factor the numerator

$5p + 20 = 5(p + 4)$
So we have $\frac{5(p + 4)}{(p + 4)(p - 4)}$

Step 6: Cancel out the common factor

Cancel out $(p + 4)$ from the numerator and the denominator (assuming $p
eq - 4$)
We get $\frac{5}{p - 4}$

Answer:

$\frac{5}{p - 4}$