Simplify:

(10^1/6 * 1000^1/2)^1/5

= (10^(1/6) * 10^(3/2) )^1/5
= ( 10^(5/3) )^1/5
= 10^(1/3 or the cube root of 10

divide write each quotient in simplest form

To simplify the expression, we can break it down step by step:

Step 1: Simplify the innermost components.

The sixth root of 10 is written as 10^(1/6).

The square root of 1000 can be simplified as follows: √1000 = √(100 * 10) = √100 * √10 = 10 * √10 = 10√10.

Now, the expression becomes:

5th root of (10^(1/6) * 10√10).

Step 2: Simplify the multiplication inside the root.

10^(1/6) * 10√10 equals 10^(1/6) * 10 * √10.

Multiplying the numbers gives us: 10^(1/6 + 1) * √10.

Simplifying further: 10^(7/6) * √10.

Step 3: Simplify the outermost root.

Taking the fifth root of the expression, we get:

(10^(7/6) * √10)^(1/5).

Using the property of exponents, we can distribute the exponent across both terms inside the parentheses:

10^((7/6) * (1/5)) * (√10)^(1/5).

Simplifying the exponents: 10^(7/30) * (√10)^(1/5).

Final Answer: The simplified expression is 10^(7/30) * (√10)^(1/5).