Final answer:
To find the value of the expression (3^2 * a^-2 / 3a^-1)^k with a = -5 and k = -2, you simplify the base and then raise it to the power of k. After simplification, the expression becomes (3/a)^(-2), which then leads to (a/3)^2 upon further simplification. Substituting a = -5 yields a final value of 25/9.
Step-by-step explanation:
The value of the expression (3^2 * a^-2 / 3a^-1)^k when a = -5 and k = -2 can be found by carefully simplifying and substituting the given values.
Firstly, simplify the base of the expression by combining like terms:
- (3^2 * a^-2 / 3a^-1) simplifies to (9 / a^2) / (3 / a).
- When you divide fractions, you multiply by the reciprocal, so the expression becomes (9 / a^2) * (a / 3).
- After cancelling out common factors, the simplified base is 3 * a^-1 or 3/a.
The next step involves raising the simplified base to the power of k, which is -2 in this case:
- (3/a)^(-2) is equivalent to (a/3)^2 according to the rules of exponents.
- Replacing a with -5, the expression now reads (-5/3)^2.
- The value of (-5/3)^2 is 25/9, because the square of a negative number is positive.
Therefore, the final value of the original expression with the given values for a and k is 25/9.