Final answer:
To simplify the expression (4³x⁻⁴y³)/(4¹²x⁻⁷y⁸)⁶, we can combine the exponents and divide the numbers. The final answer is 67,108,864x⁻⁶⁸y^-³⁰.
Step-by-step explanation:
To simplify the expression (4³x⁻⁴y³)/(4¹²x⁻⁷y⁸)⁶, we can combine the exponents using the rule (a^m)^n = a^(m*n). First, let's simplify the numerator: 4³x⁻⁴y³ = (4³)(x⁻⁴)(y³) = 64x⁻⁴y³. Next, let's simplify the denominator: 4¹²x⁻⁷y⁸ = (4¹²)(x⁻⁷)(y⁸) = 4¹²x⁻⁷y⁸. Now, we can substitute these simplified expressions back into the original expression: (64x⁻⁴y³)/(4¹²x⁻⁷y⁸)⁶ = (64x⁻⁴y³)/(4¹²x⁻⁷y⁸) × (64x⁻⁴y³)/(4¹²x⁻⁷y⁸) × (64x⁻⁴y³)/(4¹²x⁻⁷y⁸) × (64x⁻⁴y³)/(4¹²x⁻⁷y⁸) × (64x⁻⁴y³)/(4¹²x⁻⁷y⁸) × (64x⁻⁴y³)/(4¹²x⁻⁷y⁸).
When multiplying with the same base (in this case, 64x⁻⁴y³)/(4¹²x⁻⁷y⁸), we can combine the exponents using the rule (a^m)*(a^n) = a^(m+n). So, the expression simplifies to (64x⁻⁴y³)⁶ / (4¹²x⁻⁷y⁸)⁶ = 64⁶x⁻²⁴y¹⁸ / 4⁷²x⁻⁴²y⁴⁸.
Finally, we can simplify the expression further by dividing the numbers and subtracting the exponents: 64⁶ / 4⁷² = 2²⁶ = 67,108,864. And for the variables, x⁻²⁴ / x⁻⁴² = x^(⁻²⁴ - (-⁻⁴²)) = x⁻²⁴⁺⁴² = x⁻^⁶⁸, and y¹⁸ / y⁴⁸ = y^(¹⁸ - ⁴⁸) = y^-³⁰.
Therefore, the final simplified expression is 67,108,864x⁻⁶⁸y^-³⁰.