Simplify the expression using the properties of exponents. Expand any numerical portion of your answer and only include positive exponents.
(5a^3b^−1)^−3(2a^−1b)^−3
Starting with the expression (5a^3b^(-1))^(-3)(2a^(-1)b)^(-3), we can simplify it by applying the properties of exponents.
First, let's deal with the exponents. To apply the power of a power property, we multiply the exponents:
= (5^(-3)a^(3*(-3))b^(-1*(-3)))(2^(-3)a^((-1)*(-3))b^(-3))
= (5^(-3)a^(-9)b^3)(2^(-3)a^3b^(-3))
Now we can work on simplifying the numerical portion. Using the property a^(-n) = 1/a^n, we can rewrite 5^(-3) and 2^(-3) as:
= (1/5^3*a^(-9)b^3)(1/2^3*a^3b^(-3))
Now, we combine the terms with the same base. For the variables, we add the exponents, and for the numbers, we multiply them:
= (1/125*a^(-9+3)*b^(3+0))(1/8*a^3*b^(-3))
= (1/125*a^(-6)*b^3)(1/8*a^3*b^(-3))
Finally, we combine the fractions by multiplying the numerators and multiplying the denominators:
= (1*a^(-6)*b^3)/(125*8*a^3*b^(-3))
Now, let's simplify further by combining the like terms:
= (a^(-6+3)*b^(3-(-3)))/(125*8)
= (a^(-3)*b^6)/(1000)
Therefore, the simplified expression is (a^(-3)*b^6)/(1000).