Answer:
2 y^6
Step-by-step explanation:
Simplify the following:
(2 x y^2)^5/((4 x^2 y)^2 x y^2)
Hint: | Distribute exponents over products in (4 x^2 y)^2.
Multiply each exponent in 4 x^2 y by 2:
(2 x y^2)^5/(4^2 x^(2×2) y^2 x y^2)
Hint: | Multiply 2 and 2 together.
2×2 = 4:
(2 x y^2)^5/(4^2 x^4 y^2 x y^2)
Hint: | Evaluate 4^2.
4^2 = 16:
(2 x y^2)^5/(16 x^4 y^2 x y^2)
Hint: | Distribute exponents over products in (2 x y^2)^5.
Multiply each exponent in 2 x y^2 by 5:
(2^5 x^5 y^(5×2))/(16 x^4 y^2 x y^2)
Hint: | Multiply 5 and 2 together.
5×2 = 10:
(2^5 x^5 y^10)/(16 x^4 y^2 x y^2)
Hint: | Compute 2^5 by repeated squaring. For example a^7 = a a^6 = a (a^3)^2 = a (a a^2)^2.
2^5 = 2×2^4 = 2 (2^2)^2:
(2 (2^2)^2 x^5 y^10)/(16 x^4 y^2 x y^2)
Hint: | Evaluate 2^2.
2^2 = 4:
(2×4^2 x^5 y^10)/(16 x^4 y^2 x y^2)
Hint: | Evaluate 4^2.
4^2 = 16:
(2×16 x^5 y^10)/(16 x^4 y^2 x y^2)
Hint: | Multiply 2 and 16 together.
2×16 = 32:
(32 x^5 y^10)/(16 x^4 y^2 x y^2)
Hint: | In (32 x^5 y^10)/(16 x^4 y^2 x y^2), divide 32 in the numerator by 16 in the denominator.
32/16 = (16×2)/16 = 2:
(2 x^5 y^10)/(x^4 y^2 x y^2)
Hint: | For all exponents, a^n a^m = a^(n + m). Apply this to (2 x^5 y^10)/(x^4 y^2 x y^2).
Combine powers. (2 x^5 y^10)/(x^4 y^2 x y^2) = 2 x^(5 - 1 - 4) y^(10 - 2 - 2):
2 x^(5 - 1 - 4) y^(10 - 2 - 2)
Hint: | Evaluate 5 - 1 - 4.
5 - 1 - 4 = 0:
2 x^0 y^(10 - 2 - 2)
Hint: | Evaluate 10 - 2 - 2.
10 - 2 - 2 = 6:
2 x^0 y^6
Hint: | Any nonzero expression to the zero power is one.
x^0 = 1:
2×1 y^6
Hint: | Simplify the expression.
Write 2×1 y^6 as 2 y^6:
Answer: 2 y^6
