(a ^ 2 + ax)/x / ((ax)/(x ^ 2)) 7a + (2a - 7e ^ 2)/6 12/(2a - a ^ 2) - 6/a (xy + y ^ 2)/(15x) * (3x)/(x + y) 6/(x + 3) - x/((x + 3) ^ 2) / (x/(x ^ 2 - 9)) - (2x + 18)/(x + 3)

I can help you simplify these expressions. Let's go through each one step by step:

Expression 1: (a^2 + ax)/x / ((ax)/(x^2))

To simplify this, we can multiply the numerator and denominator of the first fraction by x and the denominator of the second fraction by x^2:

(a^2x + ax^2) / x * (x^2 / ax)

Now, we can cancel out common factors:

(a^2x + ax^2) * (x^2 / ax) = a^2x + x^2

So the simplified expression is: a^2x + x^2

Expression 2: 7a + (2a - 7e^2)/6

This expression is already simplified, so we leave it as it is.

Expression 3: 12/(2a - a^2) - 6/a

To simplify this expression, we need to find a common denominator:

First fraction: (12 * a) / (a * (2a - a^2))

Second fraction: (6 * (2a - a^2)) / (a * (2a - a^2))

Now, we can combine the fractions:

(12a - 6(2a - a^2)) / a(2a - a^2)

Simplifying the expression:

(12a - 12a + 6a^2) / a(2a - a^2) = 6a^2 / a(2a - a^2)

Canceling out common factors:

6a^2 / a(2a - a^2) = 6a / (2a - a^2)

So, the simplified expression is: 6a / (2a - a^2)

Expression 4: (xy + y^2)/(15x) * (3x)/(x + y)

To simplify this expression, we can multiply the numerators together and the denominators together:

(xy + y^2) * 3x / (15x * (x + y))

Now, we can cancel out common factors:

3xy(x + y) / 15x(x + y)

Simplifying:

3xy(x + y) / 15x(x + y) = 3y / 15

Reducing the fraction:

3y / 15 = y / 5

So, the simplified expression is: y / 5

Expression 5: 6/(x + 3) - x / ((x + 3)^2) / (x / (x^2 - 9)) - (2x + 18)/(x + 3)

First, let's simplify the expression x / ((x + 3)^2):

To divide fractions, we multiply the numerator by the reciprocal of the denominator:

x / ((x + 3)^2) = x * (1 / (x + 3)^2) = x / (x + 3)^2

Now, we rewrite the expression:

6/(x + 3) - x / (x + 3)^2 / (x / (x^2 - 9)) - (2x + 18)/(x + 3)

Now, let's simplify the expression (x / (x^2 - 9)):

We can factor the denominator: x^2 - 9 = (x - 3)(x + 3)

So, the expression becomes:

6/(x + 3) - x / (x + 3)^2 / (x / ((x - 3)(x + 3))) - (2x + 18)/(x + 3)

Next, let's simplify the numerator and denominator of the third fraction:

x / ((x - 3)(x + 3)) = x / ((x - 3) * (x + 3)) = x / (x^2 - 9)

Now, we rewrite the expression:

6/(x + 3) - x / (x + 3)^2 / x / (x^2 - 9) - (2x + 18)/(x + 3)

To simplify further, we can multiply the second fraction by the reciprocal of the third fraction:

6/(x + 3) - x / (x + 3)^2 * (x^2 - 9) / x - (2x + 18)/(x + 3)

Now, let's simplify the second fraction:

x / (x + 3)^2 * (x^2 - 9) / x = (x * (x - 3)(x + 3)) / ((x + 3)(x + 3)) = (x * (x - 3)(x + 3)) / (x + 3)^2

Now, we rewrite the expression:

6/(x + 3) - (x * (x - 3)(x + 3)) / (x + 3)^2 / (x) - (2x + 18)/(x + 3)

Now, we can simplify further by canceling out common factors:

6/(x + 3) - (x - 3) / (x + 3) - (2x + 18)/(x + 3)

Combining the fractions:

(6 - (x - 3) - (2x + 18)) / (x + 3) = (6 - x + 3 - 2x - 18) / (x + 3) = (-3x - 9) / (x + 3)

So, the simplified expression is: (-3x - 9) / (x + 3)