Simplify the following expression: m^{-2} - h^{-2} / m^{-2} - 2m^{-1}h^{-1} + h^{-2}

Solution:

We are asked to simplify the expression $$\frac{m^{-2} - h^{-2}}{m^{-2} - 2m^{-1}h^{-1} + h^{-2}}$$.

First, let's rewrite the expression with positive exponents. Recall that $$x^{-n} = \frac{1}{x^n}$$.

Numerator: $$m^{-2} - h^{-2} = \frac{1}{m^2} - \frac{1}{h^2} = \frac{h^2 - m^2}{m^2 h^2}$$

Denominator: $$m^{-2} - 2m^{-1}h^{-1} + h^{-2} = \frac{1}{m^2} - \frac{2}{mh} + \frac{1}{h^2}$$

To combine the terms in the denominator, find a common denominator, which is $$m^2 h^2$$.

$$ = \frac{h^2}{m^2 h^2} - \frac{2mh}{m^2 h^2} + \frac{m^2}{m^2 h^2} = \frac{h^2 - 2mh + m^2}{m^2 h^2}$$

Notice that the numerator of the denominator is a perfect square trinomial: $$m^2 - 2mh + h^2 = (m-h)^2$$. So, the denominator is $$\frac{(m-h)^2}{m^2 h^2}$$.

Now, we can write the original expression as:

$$ \frac{\frac{h^2 - m^2}{m^2 h^2}}{\frac{(m-h)^2}{m^2 h^2}} $$

To divide by a fraction, we multiply by its reciprocal:

$$ \frac{h^2 - m^2}{m^2 h^2} \times \frac{m^2 h^2}{(m-h)^2} $$

Cancel out the $$m^2 h^2$$ terms:

$$ \frac{h^2 - m^2}{(m-h)^2} $$

The numerator is a difference of squares: $$h^2 - m^2 = (h-m)(h+m)$$.

So, the expression becomes:

$$ \frac{(h-m)(h+m)}{(m-h)^2} $$

We know that $$(h-m) = -(m-h)$$. Substituting this into the expression:

$$ \frac{-(m-h)(h+m)}{(m-h)^2} $$

Cancel out one $$(m-h)$$ term from the numerator and denominator:

$$ \frac{-(h+m)}{m-h} $$

We can also write this as:

$$ \frac{h+m}{h-m} $$

Or, by multiplying the numerator and denominator by -1:

$$ \frac{-(h+m)}{-(h-m)} = \frac{-h-m}{m-h} $$

Let's double check by factoring the numerator of the denominator as $$(m-h)^2$$. The expression is $$\frac{h^2 - m^2}{(m-h)^2}$$.

Since $$h^2 - m^2 = -(m^2 - h^2) = -(m-h)(m+h)$$, we have:

$$ \frac{-(m-h)(m+h)}{(m-h)^2} = \frac{-(m+h)}{m-h} $$

This can be written as $$\frac{m+h}{h-m}$$.

Let's re-examine the denominator. The expression $$m^{-2} - 2m^{-1}h^{-1} + h^{-2}$$ can be seen as $$(\frac{1}{m} - \frac{1}{h})^2$$.

$$(\frac{1}{m} - \frac{1}{h})^2 = (\frac{h-m}{mh})^2 = \frac{(h-m)^2}{m^2h^2}$$

The numerator is $$\frac{1}{m^2} - \frac{1}{h^2} = \frac{h^2-m^2}{m^2h^2}$$.

So, we have $$\frac{\frac{h^2-m^2}{m^2h^2}}{\frac{(h-m)^2}{m^2h^2}} = \frac{h^2-m^2}{(h-m)^2}$$.

Since $$h^2 - m^2 = (h-m)(h+m)$$, the expression becomes:

$$ \frac{(h-m)(h+m)}{(h-m)^2} = \frac{h+m}{h-m} $$

This is the simplified form.

Alternatively, we can write the denominator as $$(m^{-1} - h^{-1})^2$$.

The expression is $$\frac{(m^{-1}-h^{-1})(m^{-1}+h^{-1})}{(m^{-1}-h^{-1})^2} = \frac{m^{-1}+h^{-1}}{m^{-1}-h^{-1}}$$.

$$ \frac{\frac{1}{m} + \frac{1}{h}}{\frac{1}{m} - \frac{1}{h}} = \frac{\frac{h+m}{mh}}{\frac{h-m}{mh}} = \frac{h+m}{h-m} $$

Final check:

Let $$m=1, h=2$$. Then $$m^{-2} = 1, h^{-2} = 1/4$$. $$m^{-1}=1, h^{-1}=1/2$$.

Numerator: $$1 - 1/4 = 3/4$$.

Denominator: $$1 - 2(1)(1/2) + 1/4 = 1 - 1 + 1/4 = 1/4$$.

Expression: $$(3/4) / (1/4) = 3$$.

Simplified form: $$\frac{h+m}{h-m} = \frac{2+1}{2-1} = \frac{3}{1} = 3$$. The results match.

Another way to write the denominator is $$(m^{-1}-h^{-1})^2$$.

The expression is $$\frac{(m^{-1}-h^{-1})(m^{-1}+h^{-1})}{(m^{-1}-h^{-1})^2} = \frac{m^{-1}+h^{-1}}{m^{-1}-h^{-1}}$$.

$$ \frac{\frac{1}{m} + \frac{1}{h}}{\frac{1}{m} - \frac{1}{h}} = \frac{\frac{h+m}{mh}}{\frac{h-m}{mh}} = \frac{h+m}{h-m} $$

This is equivalent to $$\frac{m+h}{h-m}$$.

It seems there might be a slight ambiguity in the interpretation of the handwritten formula. Assuming the expression is $$\frac{m^{-2} - h^{-2}}{(m^{-1} - h^{-1})^2}$$ as suggested by the denominator structure.

Let's re-evaluate the denominator $$m^{-2} - 2m^{-1}h^{-1} + h^{-2}$$. This is indeed $$(m^{-1} - h^{-1})^2$$.

So, the expression is $$\frac{m^{-2} - h^{-2}}{(m^{-1} - h^{-1})^2}$$.

We can factor the numerator as a difference of squares: $$m^{-2} - h^{-2} = (m^{-1} - h^{-1})(m^{-1} + h^{-1})$$.

So, the expression becomes:

$$ \frac{(m^{-1} - h^{-1})(m^{-1} + h^{-1})}{(m^{-1} - h^{-1})^2} $$

Cancel out one term of $$(m^{-1} - h^{-1})$$ from the numerator and denominator:

$$ \frac{m^{-1} + h^{-1}}{m^{-1} - h^{-1}} $$

Now, substitute back $$m^{-1} = \frac{1}{m}$$ and $$h^{-1} = \frac{1}{h}$$:

$$ \frac{\frac{1}{m} + \frac{1}{h}}{\frac{1}{m} - \frac{1}{h}} $$

Find a common denominator for the numerator and the denominator:

Numerator: $$\frac{1}{m} + \frac{1}{h} = \frac{h}{mh} + \frac{m}{mh} = \frac{h+m}{mh}$$

Denominator: $$\frac{1}{m} - \frac{1}{h} = \frac{h}{mh} - \frac{m}{mh} = \frac{h-m}{mh}$$

Now, divide the numerator by the denominator:

$$ \frac{\frac{h+m}{mh}}{\frac{h-m}{mh}} $$

Multiply the numerator by the reciprocal of the denominator:

$$ \frac{h+m}{mh} \times \frac{mh}{h-m} $$

Cancel out the $$mh$$ terms:

$$ \frac{h+m}{h-m} $$

This can also be written as $$\frac{m+h}{h-m}$$ or $$-\frac{m+h}{m-h}$$.

Ответ: $$\frac{m+h}{h-m}$$.