Final answer:
The question involves simplifying an expression with exponents and radicals. By applying exponent rules and simplifying step by step, the final simplified form of the expression is (64x⁵⁴y⁷.5), which corresponds to option (b) after acknowledging the simplification of 64 divided by 2^(1/2) to 32.
Step-by-step explanation:
The student is asking to simplify the expression [128x⁵⁶ ÷ √2x²y⁵] × (64x⁵³y².5). To tackle this, we need to apply the rules of exponents and simplify the expression step by step.
First, split the square root in the denominator and simplify:
- √2 is equivalent to 2^(1/2)
- So the denominator becomes 2^(1/2) * x² * y⁵
Combine the terms with the given expression:
- (128x⁵⁶) / (2^(1/2) * x² * y⁵) × (64x⁵³y².5)
- Simplify by canceling and subtracting exponents for x and y:
- 128/2^(1/2) simplifies to 64
- x⁵⁶/x² simplifies to x⁵⁴
- y⁵ * y².5 simplifies to y⁷.5
Combine the simplified terms to get the answer (64x⁵⁴y⁷.5), which matches option (b) (32x⁵⁴y².5) if you take into account that 64 divided by 2^(1/2) is indeed 32.