Final answer:
The expression ∛81x^7y^9z^5 simplifies to 3x^2y^3z∛xz^2 by applying the cube root to each term and using the rule that the exponents inside the radical are divided by 3.
Step-by-step explanation:
To simplify the radical expression ∛81x^7y^9z^5, we need to break it down into its prime factors and apply the cube root to each of them.
Firstly, 81 is 3 to the power of 4 (3^4), so its cube root is 3. Since we have variables with exponents in the expression, we apply the rule of cubing exponentials, which states that we divide the exponent by 3. For x^7, we take the cube root of x^3 and get x^2 outside the radical, with 1 power of x remaining inside. The full cube root of y^9 is simply y^3. Lastly, for z^5, the cube root of z^3 gives us z outside the radical, with z^2 remaining inside.
Therefore, the simplified form of the radical expression is 3x^2y^3z∛xz^2. We can check that this answer is reasonable by ensuring that we have correctly applied the rule of cubing exponentials and reduced the original expression without losing any terms.