Simplify the expression using the properties of exponents. Expand any numerical portion of your answer and only include positive exponents.
(4m^3n^−2)^−3/(4m^−1n)^−2
To simplify this expression, we first need to apply the properties of exponents.
First, we can rewrite the expression as follows:
(4m^3n^(-2))^(-3)/(4m^(-1)n)^(-2)
Now, let's simplify the exponents within the parentheses:
(4^(-3) * m^(3*(-3)) * n^((-2)*(-3)))/(4^(-2) * m^((-1)*(-2)) * n^(-2))
Next, simplify the constants outside of the parentheses:
(1/4^3 * m^(-9) * n^6)/(1/4^2 * m^2 * n^(-2))
Now, let's simplify the exponent expressions:
For the m terms:
1/4^3 is 1/64
m^(-9) can be written as 1/m^9
m^2 is m^2
For the n terms:
n^6 is n^6
1/4^2 is 1/16
1/n^(-2) is n^2
Now, we can simplify the whole expression:
(1/64 * 1/m^9 * n^6)/(1/16 * m^2 * n^2)
Simplifying further, we divide the fractions:
(1 * 16 * m^2 * n^2)/(64 * m^9 * n^6)
Now, we can cancel out common factors:
1 * 16 = 16
m^2/m^9 = 1/m^7
n^2/n^6 = 1/n^4
Our final answer is:
16/n^4m^7