To solve the expression (√6)^(8x) * (216x^3), we start by analyzing each part of the expression.
First, let’s simplify (√6)^(8x). We know that the square root of a number can be expressed as that number raised to the power of 1/2. Thus, we can rewrite it as:
(√6)^(8x) = (6^(1/2))^(8x) = 6^(4x).
Now, we can rewrite the full expression as:
6^(4x) * (216x^3).
Next, we notice that 216 can be expressed as:
216 = 6^3.
Now we can substitute this back into our expression:
6^(4x) * (6^3 * x^3) = 6^(4x + 3) * x^3.
This is the simplified form of the original expression. Therefore, the final result is:
6^(4x + 3) * x^3.
This shows how exponents can be managed through addition when multiplying like bases. The terms involving the variable x are untouched as they multiply directly in the overall expression.