What is the second simplified form of these two problems if there is any other way?
3. There are many ways to simplify exponents. Provide two different ways to simplify each of the problems below. Please write all final answers using positive exponents.
problem 1: (-4m^2 n^2)^3 * (2m^3 n)^-1
simplified 1: -(32n^5/m^9)
simplified 2: ?
problem 2: (x^-(1/2) y^3)^-2/ xy^(1/3) * x^2 y^-4
simplified 1: 1/x^2y^(7/2)
simplified 2: ?
To find the second simplified form of each problem, let's break down the process step by step:
Problem 1: (-4m^2n^2)^3 * (2m^3n)^(-1)
Step 1: Simplify the exponents inside the parentheses.
(-4^3 * m^(2*3) * n^(2*3)) * (2^(-1) * m^(3*(-1)) * n^(-1))
Step 2: Simplify the exponents using the power of a power and power of a product rules.
(-64m^6n^6) * (2^(-1) * m^(-3) * n^(-1))
Step 3: Simplify the negative exponents by moving them to the denominator.
(-64m^6n^6) / (2 * m^3 * n)
Step 4: Combine like terms and simplify.
-32m^3n^5
So, the second simplified form of problem 1 is -32m^3n^5.
Problem 2: (x^(-1/2)y^3)^(-2) / (xy^(1/3)) * x^2 * y^(-4)
Step 1: Simplify the exponents inside the parentheses.
(x^((-1/2)*(-2)) * y^(3*(-2))) / (xy^(1/3)) * x^2 * y^(-4)
Step 2: Simplify the exponents using the power of a power and power of a product rules.
(x^1/ y^-6) / (xy^(1/3)) * x^2 * y^(-4)
Step 3: Simplify the negative exponents by moving them to the denominator.
(x / y^6) / (xy^(1/3)) * x^2 * y^(-4)
Step 4: Combine like terms and simplify.
(x^(1+2) / y^(6+(1/3-4))) * 1
Step 5: Simplify the exponent of y in the denominator.
(x^3 / y^(18/3-12))
Step 6: Simplify the exponent of y.
(x^3 / y^2)
So, the second simplified form of problem 2 is x^3 / y^2.