Simplify the expression using the properties of exponents. Expand any numerical portion of your answer and only include positive exponents.
(4x^2y^−1)^−2/(3x^−2y)^−3
To simplify the expression using the properties of exponents, we need to apply the power rule for exponents.
According to the power rule, when we raise a power to another power, we need to multiply the exponents.
First, let's apply the power rule to each term in the numerator and denominator:
In the numerator: (4x^2y^−1)^−2
= 4^−2 * (x^2)^−2 * (y^−1)^−2
= 1/(4^2 * x^4 * y^−2) (since negative exponents indicate taking the reciprocal)
In the denominator: (3x^−2y)^−3
= (3^−1 * x^−2 * y)^−3
= (1/3 * x^−2 * y)^−3
= 1/(1/3)^3 * (x^−2)^−3 * (y)^−3
= 1/(1/27 * x^6 * y^−3)
= 27/x^6y^3 (since taking the reciprocal of 1/27 is 27)
Now, let's simplify the expression by dividing the terms in the numerator by the terms in the denominator:
(1/(4^2 * x^4 * y^−2)) / (27/x^6y^3)
= 1/(4^2 * x^4 * y^−2) * (x^6y^3)/27
= (x^6 * y^3) / (4^2 * x^4 * y^−2 * 27)
= x^(6-4) * y^(3+2) / (4^2 * 27)
= x^2 * y^5 / (16 * 27)
Finally, we can expand the numerical portion:
= x^2 * y^5 / 432