Simplify and write the answer using positive exponents :
1. (3^3)(3^-6)
2. S^2 t^-1 / (s^4 t^4)^1/2
3. [(2/3)^2]^-3
To simplify and write the answers using positive exponents, we can follow these steps:
1. (3^3)(3^-6):
Using the rule for multiplying exponents with the same base, we add the exponents together. In this case, 3 + (-6) = -3. So, the expression becomes 3^-3.
Answer: 3^-3
2. S^2 t^-1 / (s^4 t^4)^1/2:
First, let's simplify the expression inside the parentheses. Using the rule for exponents, we multiply the exponents inside the parentheses with the exponent outside the parentheses. (s^4)^1/2 = s^2 and (t^4)^1/2 = t^2.
Now, the expression becomes S^2 t^-1 / (s^2 t^2):
Using the rule for dividing exponents with the same base, we subtract the exponents. In this case, 2 - (-1) = 3 for 'S', and 2 - 1 = 1 for 't'. So, the expression becomes S^3 t^1.
Answer: S^3 t
3. [(2/3)^2]^-3:
First, let's simplify the expression inside the square brackets. (2/3)^2 = 4/9.
Now, the expression becomes (4/9)^-3:
Using the rule for negative exponents, we move the expression to the denominator by changing the sign of the exponent. So, it becomes 1 / (4/9)^3.
Using the rule for exponents, we raise the numerator and denominator to the power 3. (1^3)/(4/9)^3 = 1 / (64/729).
To divide fractions, we multiply by the reciprocal. 1 / (64/729) = 1 * (729/64).
Simplifying, we get 729/64.
Answer: 729/64