Answer: To simplify the expression, let's first combine the fractions in the numerator and denominator separately:
Numerator:
(a/m^2) + (a^2/m^3)
Denominator:
(m^2/a^2) + (m/a)
Now, we can rewrite the original expression as:
[(a/m^2) + (a^2/m^3)] / [(m^2/a^2) + (m/a)]
To simplify further, we need to find the common denominator for the terms in the numerator and denominator.
The common denominator for the numerator terms is m^3, and for the denominator terms is a^2.
Now, rewrite the expression using the common denominators:
(a * m / m^3 + a^2) / (m^2 + m^3 / a^2)
Next, let's combine the terms in the numerator:
[(a * m + a^2 * m^3) / m^3] / (m^2 + m^3 / a^2)
Now, invert the denominator to change the division into multiplication:
[(a * m + a^2 * m^3) / m^3] * (a^2 / (m^2 + m^3 / a^2))
Now, let's simplify further by canceling out common factors:
[(a * m + a^2 * m^3) / m^3] * (a^2 / (m^2 + (m^3 * a) / a^2))
Now, multiply the numerators and denominators:
(a^3 * m + a^4 * m^3) / (m^3 * (m^2 + m^3 * a / a^2))
Next, distribute the m^3 term inside the bracket:
(a^3 * m + a^4 * m^3) / (m^3 * (m^2 + m^2))
Now, simplify the expression further:
(a^3 * m + a^4 * m^3) / (m^3 * 2 * m^2)
Now, cancel out the common factor of m^2 in the numerator and denominator:
(a^3 + a^4 * m^2) / (2 * m)
The simplified expression is:
(a^3 + a^4 * m^2) / (2 * m)