Simplify:
5th roof of (6th root of 10 * sqaure root of 1000)
or:
(10^1/6 * 1000^1/2)^1/5
(^ another way of writing it)
(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).