Simplify

Problem

((32*x5*y(−3/2))(2/5))/((x(5/3)*y(2/3))(3/5))

Solution

1. Apply the power of a product rule to the numerator by raising each factor inside the parentheses to the power of 2/5

(32*x5*y(−3/2))(2/5)=32(2/5)⋅(x5)(2/5)⋅(y(−3/2))(2/5)

2. Simplify the numerator by calculating the numerical value and multiplying the exponents.

32(2/5)=(2)(2/5)=2=4

(x5)(2/5)=x(5⋅2/5)=x2

(y(−3/2))(2/5)=y(−3/2⋅2/5)=y(−3/5)

3. Apply the power of a product rule to the denominator by raising each factor inside the parentheses to the power of 3/5

(x(5/3)*y(2/3))(3/5)=(x(5/3))(3/5)⋅(y(2/3))(3/5)

4. Simplify the denominator by multiplying the exponents.

(x(5/3))(3/5)=x(5/3⋅3/5)=x1=x

(y(2/3))(3/5)=y(2/3⋅3/5)=y(2/5)

5. Combine the simplified parts into a single fraction.

(4*x2*y(−3/5))/(x*y(2/5))

6. Apply the quotient rule for exponents by subtracting the exponents of like bases.

4⋅x(2−1)⋅y(−3/5−2/5)

4⋅x1⋅y(−5/5)

4*x*y(−1)

7. Rewrite with positive exponents by moving the factor with the negative exponent to the denominator.

(4*x)/y

Final Answer

((32*x5*y(−3/2))(2/5))/((x(5/3)*y(2/3))(3/5))=(4*x)/y

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