Sure, here are the steps to simplify the given fraction:
Step 1:
We start with the given fraction ((5s^3n^2)^-1) / (4s^3).
Step 2:
First, let's simplify the numerator. For any nonzero number a and integer m, (a^-m) equals 1/a^m. This is one of our rules of exponents, which states that when a base is raised to a negative exponent, we can write it as the reciprocal of the base raised to the positive exponent.
So, (5s^3n^2)^-1 can be rewritten as 1/(5s^3n^2).
Step 3:
Now our expression looks like: 1/(5s^3n^2) / 4s^3.
Step 4:
Recall from basic fraction operations that when we divide fractions, we actually multiply the first fraction by the reciprocal of the second fraction. Treat 4s^3 as 4s^3/1 so that it resembles the fraction form.
So, 1/(5s^3n^2) / 4s^3 can be rewritten as 1/(5s^3n^2) * 1/(4s^3).
Step 5:
When multiplying fractions, we multiply the numerators and the denominators, i.e., a/b * c/d = (a*c) / (b*d).
Therefore, 1/(5s^3n^2) * 1/(4s^3) becomes 1/[(5s^3n^2)*(4s^3)].
Step 6:
Now, let's multiply the denominators. Recall that when multiplying powers with similar bases, you add the exponents. Hence the s^3 from 5s^3n^2 and the s^3 from 4s^3 will become s^6.
So, 1/[(5s^3n^2)*(4s^3)] becomes 1/(20*s^6*n^2).
That's your final expression, in simplest form, with positive exponents.