Questions: Simplify the expression. Assume that all variables are nonnegative real numbers. ∛(81 a) ∛(81 a) = (Simplify your answer. Use positive exponents only. Type an exact answer, using radicals as needed.)

Solution Steps

To simplify the expression \(\sqrt[3]{81a}\), we need to break down the number 81 into its prime factors and then apply the cube root to both the numerical and variable parts separately.

1. Factorize 81 into its prime factors.
2. Apply the cube root to the numerical part.
3. Apply the cube root to the variable part.

The number \( 81 \) can be expressed as \( 3^4 \). Therefore, we can rewrite the expression as: \[ \sqrt[3]{81a} = \sqrt[3]{3^4 a} \]

Using the property of cube roots, we can separate the numerical and variable parts: \[ \sqrt[3]{3^4 a} = \sqrt[3]{3^4} \cdot \sqrt[3]{a} \] The cube root of \( 3^4 \) can be simplified as follows: \[ \sqrt[3]{3^4} = 3^{4/3} = 3 \cdot 3^{1/3} \]

Now, we can combine the results from the previous steps: \[ \sqrt[3]{81a} = 3 \cdot 3^{1/3} \cdot a^{1/3} \] This can be expressed as: \[ 3 \cdot 3^{1/3} \cdot a^{1/3} \]

Final Answer