(a/b^2-1/a):(1/b+1/a)

To simplify the expression \((\frac{a}{b^2} - \frac{1}{a}) : \left(\frac{1}{b} + \frac{1}{a}\right)\), we can start by rewriting it in a more manageable form. We'll convert it to a fraction:

\[ \frac{\frac{a}{b^2} - \frac{1}{a}}{\frac{1}{b} + \frac{1}{a}} \]

Next, we'll simplify both the numerator and the denominator.

Step 1: Simplify the numerator \(\frac{a}{b^2} - \frac{1}{a}\)

To combine the terms in the numerator, we need a common denominator, which would be \(ab^2\):

\[ \frac{a^2}{ab^2} - \frac{b^2}{ab^2} = \frac{a^2 - b^2}{ab^2} \]

Step 2: Simplify the denominator \(\frac{1}{b} + \frac{1}{a}\)

Again, we need a common denominator for the terms in the denominator, which would be \(ab\):

\[ \frac{a}{ab} + \frac{b}{ab} = \frac{a + b}{ab} \]

Step 3: Rewrite the overall fraction

Now, we replace the numerator and the denominator in the original expression:

\[ \frac{\frac{a^2 - b^2}{ab^2}}{\frac{a + b}{ab}} = \frac{a^2 - b^2}{ab^2} \cdot \frac{ab}{a + b} \]

Step 4: Simplify the fraction

When multiplying, we cancel \(ab\) in the numerator and denominator:

\[ \frac{(a^2 - b^2)}{b(a + b)} \]

Step 5: Factor \(a^2 - b^2\)

The numerator can be factored:

\[ a^2 - b^2 = (a - b)(a + b) \]

So our expression now looks like this:

\[ \frac{(a - b)(a + b)}{b(a + b)} \]

Step 6: Cancel out common factors

Provided \(a + b \neq 0\) (which we assume), we can cancel \(a + b\) from the numerator and denominator:

\[ \frac{a - b}{b} \]

Thus, the final simplified form of the original expression is:

\[ \frac{a - b}{b} \]