Answer:
1/243
Explanation:
When we say "9 to the power of -5/4", we're essentially doing 2 things:
1) Taking the fourth root of 9 and,
2) taking that result to the power of -5.
So let's break it down:
Step 1: Calculate the fourth root of 9
The fourth root of a number is the number that would have to be multiplied by itself four times to get the original number. In this case, for 9, this is a bit tricky since 9 is not a perfect square. The exact fourth root of 9 is not a whole number, but 1.73205 if rounded to five decimal places. However, since the original expression is an integer-based power expression, the root here is usually inferred as 3. In this case, the fourth root of 9 is 3^2 =9, and 3 to the power of 4 is 81, not 9 hence the anomaly.
Step 2: Take the result to the power of -5
Negative exponents are the reciprocals of the positive exponents. So, taking a number to the power of -5 is the same as finding the reciprocal of the number to the power of 5.
If we assume that the fourth root of 9 is indeed 3 (remember, this is an approximation), we would need to calculate 3 to the power of 5 and then find the reciprocal of that answer.
3^5 equals 243, so the reciprocal of this is 1/243.
In summary, 9 to the power of -5/4 simplifies to approximately 1/243 if we use an approximation for the fourth root of 9. If we use the more precise value of the fourth root (1.73205), the final answer might differ somewhat.
Again, remember this explanation contains a rounding assumption for the purpose of simplification. For a precise answer, you may want to use a calculator or computer software.