Express in simplest radical form. Do NOT give decimal approximations.

1. n^3/4 p^5/4

2. (3r)^2/5 s^3/5

3. (1/32)^0.2

4. 140^-1/2

fractional exponents are roots, where the radical index is the denominator. So, you have

n^3/4 p^5/4
= (n^3 p^5)^1/4
= ∜(n^3 p^5)
or, if you want proper fractions for the roots,
= p∜(n^3 p)

To express the given expressions in the simplest radical form, we need to simplify the exponents and simplify any radicals if possible. Let's go through each expression:

1. n^(3/4) p^(5/4):
To simplify the exponents, we can rewrite the expression as a radical. The numerator of the exponent represents the power of n, and the denominator represents the root. So, n^(3/4) can be written as the fourth root of n^3, and p^(5/4) can be written as the fourth root of p^5. Therefore, the expression becomes ∛(n^3) ∛(p^5), which can also be written as ∛(n^3) * ∛(p^5).

2. (3r)^(2/5) s^(3/5):
Similarly, we can rewrite the expression as radicals. The numerator of the exponent represents the power of (3r), and the denominator represents the root. So, (3r)^(2/5) can be written as the fifth root of (3r)^2, and s^(3/5) can be written as the fifth root of s^3. Therefore, the expression becomes ∜((3r)^2) ∜(s^3), which simplifies to 2∜(3r) * ∜(s^3).

3. (1/32)^0.2:
To simplify the expression, we evaluate the exponent. Since the exponent is 0.2, we can write it as a fractional exponent. 0.2 can be expressed as 1/5. Therefore, (1/32)^0.2 is equal to (1/32)^(1/5). To combine the bases of the fraction, we keep the denominator as it is since it represents the root, and write the numerator as the power. So, (1/32)^(1/5) becomes the fifth root of 1/32.

4. 140^(-1/2):
To simplify this expression, we can rewrite it as a fraction with a positive exponent. 140^(-1/2) can be rewritten as 1/√(140). To further simplify, we find the square root of 140.

Therefore, the simplest radical forms are:

1. ∛(n^3) * ∛(p^5)
2. 2∜(3r) * ∜(s^3)
3. (1/32)^(1/5)
4. 1/√(140)