multiply and simplify by factoring. Assume that all expressions under the radicals represent nonnegative numbers.

^4 sq. rt. (810x^8y10)=
show steps if possible.

To simplify the expression ^4√(810x^8y^10), we can follow these steps:

Step 1: Find the prime factorization of the number under the radical.

The prime factorization of 810 can be found by dividing it by smaller prime numbers:

810 ÷ 2 = 405
405 ÷ 3 = 135
135 ÷ 3 = 45
45 ÷ 3 = 15
15 ÷ 5 = 3

So, the prime factorization of 810 is 2^1 × 3^4 × 5^1.

Step 2: Rewrite the expression with the prime factorization.

^4√(810x^8y^10) = ^4√(2^1 × 3^4 × 5^1 × x^8 × y^10)

Step 3: Group the factors into sets of four, and write each group as a separate radical.

^4√(2^1 × 3^4 × 5^1 × x^8 × y^10) = ^4√(2^1 × 3^4) × ^4√(5^1 × x^8) × ^4√(y^10)

Step 4: Simplify each of the separate radicals.

^4√(2^1 × 3^4) = ^4√(2) × ^4√(3^4) = √2 × √3 = √6
^4√(5^1 × x^8) = ^4√(5) × ^4√(x^8) = √5 × √(x^2)^2 = √5 × x^2 = x^2√5
^4√(y^10) = ^4√(y^8 × y^2) = √(y^2)^4 × √(y^2) = y^2 × y = y^3

Step 5: Combine the simplified radicals.

^4√(810x^8y^10) = √6 × x^2√5 × y^3

Therefore, the simplified expression is √6x^2√5y^3.