what is two ways to simplify the two problems?
problem 1: (-4m^2 n^2)^3 * (2m^3 n)^-1
solution 1: -(32n^5/m^9)
Solution 2: ?
problem 2: (x^-(1/2) y^3)^-2/ xy^(1/3) * x^2 y^-4
solution 1: 1/x^2y^(7/2)
solution 2: ?
To simplify the given problems, we'll apply the rules of exponents and algebraic manipulation.
Problem 1: (-4m^2 n^2)^3 * (2m^3 n)^-1
Solution 1:
Step 1: Simplify the exponents within each parenthesis:
(-4)^3 * (m^2)^3 * (n^2)^3 * (2m^3)^-1 * (n)^-1
-64 * m^6 * n^6 * 1/(2m^3) * 1/n
Step 2: Combine similar terms:
-64m^6n^6 / (2m^3n)
Step 3: Simplify further:
-32m^(6-3) * n^(6-1)
-32m^3 * n^5
Solution 1: -32m^3n^5
Solution 2:
Alternatively, you can expand the negative exponent and simplify:
(-4m^2 n^2)^3 * (2m^3 n)^-1
(-4^3 * m^(2*3) * n^(2*3)) / (2 * m^-3 * n^-1)
(-64 * m^6 * n^6) / (2 * (1/m^3) * (1/n))
(-32 * m^6 * n^6) / (m^3 * n)
(-32m^(6-3) * n^(6-1))
-32m^3 * n^5
Solution 2: -32m^3n^5
Problem 2: (x^-(1/2) y^3)^-2/ xy^(1/3) * x^2 y^-4
Solution 1:
Step 1: Apply the negative exponent to eliminate it:
(x^(1/2) y^-3)^2 / xy^(1/3) * x^2 y^-4
(x^(1/2 * 2) * y^(-3 * 2)) / (x * y^(1/3)) * (x^2 * y^-4)
(x * y^-6) / (x * y^(1/3)) * (x^2 * y^-4)
Step 2: Simplify by combining the same variables:
1 / y^(1/3 - 6 + 4) * x^(1+2)
1 / y^(-17/3) * x^3
1/x^3 * y^(17/3)
Solution 1: 1/x^3 * y^(17/3)
Solution 2:
Another way to simplify is by applying the laws of exponents:
(x^-(1/2) y^3)^-2/ xy^(1/3) * x^2 y^-4
(1/(x^(1/2) y^3))^2 / (xy^(1/3)) * (x^2 * y^-4)
(1/x^(1/2 * 2) * y^(3 * 2)) / xy^(1/3) * (x^2 * y^-4)
(1/x * y^6) / xy^(1/3) * (x^2 * y^-4)
(1/x * y^6) / (xy^(1/3) * x^2 * y^-4)
1 / (x * x^2) * y^(6 - 1/3 - 4)
1/(x^3) * y^(17/3)
Solution 2: 1/x^3 * y^(17/3)
Both solutions for each problem are equivalent, so you can choose either one.