Simplify each of the following problems. Write all answers with positive exponents and simplified radical form.

1:
(64m^2)^3/2

2:
(m x m^-2 n^1/3)^2

3:
a^5/6 b^2/6

4:
4x^-2/-4
___
8x^1/3

(64m^2)^3/2 = 64^3/2 * (m^2)^3/2 = 512m^3

(m x m^-2 n^1/3)^2 = (m^-1 n^1/3)^2 = m^-2 n^2/3
a^5/6 b^2/6 = (a^5 b^2)^1/6

(4x^-2)/-4
------------ = (-x^-2)/(8x^1/3) = -1/(8 x^(7/3) )
8x^1/3

To simplify each of the given problems, we will apply the rules of exponents and simplify the expressions step by step.

1. To simplify (64m^2)^(3/2):
Taking the exponent (3/2) to each term inside the parentheses:
(64^(3/2))(m^(2(3/2)))
Simplifying the expression inside the parentheses:
(64^(3/2))(m^(6/2))
Using the rule that (a^m)^n = a^(m*n):
(64^(3/2))(m^3)
Evaluating 64^(3/2) as the square root of 64 cubed:
(8^3)(m^3)
Simplifying 8^3 = 8 x 8 x 8 = 512:
512m^3

2. To simplify (m * m^-2 * n^(1/3))^2:
Simplifying the expression inside the parentheses:
(m * m^-2 * n^(1/3))(m * m^-2 * n^(1/3))
Multiplying all the terms inside the parentheses:
(m^1 * m^-2 * n^(1/3))^2
Applying the rule m^a * m^b = m^(a+b):
m^(1-2) * n^(1/3 * 1/3)
Simplifying exponents:
m^(-1) * n^(1/9)
Rewriting m^(-1) as 1/m^1:
1/m * n^(1/9)
Simplified form: n^(1/9)/m

3. To simplify a^(5/6) * b^(2/6):
Simplifying the exponents:
a^(5/6) * b^(1/3)
Rewriting a^(5/6) as the sixth root of a to the power of 5:
∛(a^5) * ∛(b^2)
Combining under a single radical:
∛(a^5 * b^2)
Simplified form: ∛(a^5 * b^2)

4. To simplify (4x^-2)/(-4)/(8x^(1/3)):
Simplifying the numerator (4x^-2):
4/x^2
Simplifying the denominator (-4)/(8x^(1/3)):
-1/(2x^(1/3))
Combining the terms:
(4/x^2) / (-1/(2x^(1/3)))
Reciprocating the denominator and simplifying:
(4/x^2) * (-2x^(1/3))
Multiplying the terms:
-8x^(1/3)/x^2
Applying the rule x^a / x^b = x^(a-b):
-8x^(1/3 - 2)
Simplifying exponents:
-8x^(-5/3)
Rewriting x^(-5/3) as 1/x^(5/3):
-8/x^(5/3)
Simplified form: -8/x^(5/3)