1. Simplify with positive exponents (evaluate where possible AFTER simplifying)
a) square root (12s^-3t^4)(27s^2t^-5) over 9st^-7
To simplify this expression, we can start by simplifying the square root and then combining like terms.
First, let's simplify the square root term inside the parentheses:
√(12s^-3t^4) = √(12) * √(s^-3) * √(t^4) = 2√3 * (1/s^3) * t^2 = 2√3t^2/s^3
Similarly, simplify the square root term outside the parentheses:
√(27s^2t^-5) = √(27) * √(s^2) * √(t^-5) = 3√3 * s * (1/t^2) = 3√3s/t^2
Now, we can rewrite the entire expression:
(2√3t^2/s^3) * (3√3s/t^2) over 9st^-7
Multiply the numerators together and the denominators together:
(2√3 * 3√3 * t^2 * s) / (s^3 * t^2 * 9 * s * t^-7)
Simplify the numbers and simplify the variables by adding the exponents:
6 * 3 * t^2 * s / (9 * s^4 * t^-5)
Combine the like terms:
18t^2s / (9s^4t^-5)
Simplify the coefficients:
2t^2s / s^4t^-5
Now, combine the variables by subtracting the exponents:
2t^2s / s^4 * t^5
Simplify:
2t^2s / s^4t^5
Finally, cancel out the common factors:
2 / s^3t^3