What is a simpler form of the radical expression? ^4sqrt 81x^20 y^8 A, 3|x^5|y^2 B. 9|x^25|y^4 C. 9x^25|y^4| D. 3x^5|y^2|
To simplify the given radical expression, ^4sqrt 81x^20 y^8, we need to factorize the radicand (the number under the radical sign) and then simplify it using the properties of radicals.
Let's break down the given expression step by step:
1. Start with ^4sqrt 81x^20 y^8.
2. To simplify the expression inside the radical, let's factorize 81x^20 y^8.
- 81 can be written as 3^4.
- x^20 can be written as (x^10)^2.
- y^8 can be written as (y^4)^2.
3. Combining the factorizations, we have ^4sqrt (3^4 * (x^10)^2 * (y^4)^2).
4. Apply the property of radicals, which states that ^n√(a * b) = ^n√a * ^n√b. This allows us to separate the factors inside the radical.
- ^4sqrt (3^4 * (x^10)^2 * (y^4)^2) = (^4sqrt 3^4) * (^4sqrt (x^10)^2) * (^4sqrt (y^4)^2).
5. Simplify each factor inside the radical:
- ^4sqrt 3^4 = 3.
- ^4sqrt (x^10)^2 = (x^10)^(2/4) = x^5.
- ^4sqrt (y^4)^2 = (y^4)^(2/4) = y^2.
6. Combine the simplified factors:
- 3 * x^5 * y^2.
Therefore, a simpler form of the radical expression ^4sqrt 81x^20 y^8 is 3x^5y^2, which corresponds to option A.