Simplify:
[(4x^3y)(4^2xy)^-1]/(4xy^2)
A. x /16y^2
B. 16x/y^2
C. x^2/16y
D. y/16x^2
To simplify the given expression, let's break it down step by step.
1. Start with the numerator: [(4x^3y)(4^2xy)^-1]
- Simplify the exponent in the denominator: (4^2xy)^-1 means 1/(4^2xy).
- Recall that a negative exponent can be converted to a positive exponent by taking the reciprocal of the base.
2. Multiply the numerator and denominator: [(4x^3y)(1/(4^2xy))]/(4xy^2)
- Multiply the numerators and denominators together.
3. Combine like terms and simplify:
- First, simplify the denominator: 4xy^2.
- Next, multiply the remaining terms in the numerator: (4x^3y) * (1/(4^2xy)), which simplifies to (4x^2/x^2) * (1/y).
- Simplify further: 4x^2/x^2 = 4, and (1/y) is equivalent to 1 * (1/y) = 1/y.
4. Final simplified expression:
- The simplified expression is 4 / (4xy^2y), which can be further simplified to 1 / (xy).
- Multiplying the numerator and the denominator by (1/x) gives us (1 / (xy)) * (1/x), which simplifies to 1 / (x * xy).
- Further simplifying: (1 / (x * xy)) is equivalent to (1 / (x^2y)).
Therefore, the correct answer is C. x^2/16y.