Math Snap

STEP 1

Assumptions1. m and n are real numbers. n is not equal to0, as it is in the denominator

STEP 2

First, we simplify the term $(-m)^{0}$.
Any non-zero number raised to the power of0 is1.
$$(-m)^{0} =1$$

STEP 3

Now, we substitute $(-m)^{0}$ with1 in the original expression.
$$\left[\left(\frac{m}{n}\right)^{2}-1\right] \div\left(m n^{-1}+1\right)$$

STEP 4

Next, we simplify the term $\left(\frac{m}{n}\right)^{2}$.
This is equivalent to $\frac{m^2}{n^2}$.
$$\left[\frac{m^2}{n^2}-1\right] \div\left(m n^{-1}+1\right)$$

STEP 5

We simplify the term $m n^{-1}$.
This is equivalent to $\frac{m}{n}$.
$$\left[\frac{m^2}{n^2}-1\right] \div\left(\frac{m}{n}+1\right)$$

STEP 6

Now, we simplify the expression by subtracting1 from $\frac{m^2}{n^2}$.
This is equivalent to $\frac{m^2-n^2}{n^2}$.
$$\frac{m^2-n^2}{n^2} \div\left(\frac{m}{n}+1\right)$$

STEP 7

Next, we rewrite the division as a multiplication by the reciprocal.
$$\frac{m^2-n^2}{n^2} \times\left(\frac{n}{m}+1\right)^{-1}$$

STEP 8

We simplify the term $\left(\frac{n}{m}+1\right)^{-1}$.
This is equivalent to $\frac{m}{m+n}$.
$$\frac{m^2-n^2}{n^2} \times\frac{m}{m+n}$$

Finally, we multiply the two fractions together.
$$\frac{m^2-n^2}{n^2} \times\frac{m}{m+n} = \frac{m(m^2-n^2)}{n^2(m+n)}$$The simplified expression is $\frac{m(m^2-n^2)}{n^2(m+n)}$.