The power $3^{-3}$ equals $\frac{1}{27}$. Which expression is equivalent to $3^{-3}$?
$\frac{1}{3^2}$
$\frac{1}{3^5}$
$\frac{1}{9^3}$
$\frac{1}{3^5}$
Answer (1)
Calculate the value of 3 − 3 which equals 27 1 .
Evaluate each of the given expressions: 3 2 1 = 9 1 , 3 5 1 = 243 1 , 9 3 1 = 729 1 .
Compare the value of 3 − 3 with the values of the given expressions.
None of the given expressions are equivalent to 3 − 3 . There is likely an error in the question or the options.
Explanation
Understanding the problem We are given that 3 − 3 = 27 1 . We need to find which of the given expressions is equivalent to 3 − 3 .
Evaluating the expressions Let's evaluate each of the given expressions:
3 2 1 = 9 1
3 5 1 = 243 1
9 3 1 = ( 3 2 ) 3 1 = 3 6 1 = 729 1
3 5 1 = 243 1
Comparing the values Now, let's compare the values we calculated with 3 − 3 = 27 1 .
9 1 = 27 1 243 1 = 27 1 729 1 = 27 1
Re-evaluating the problem However, we made a mistake in the original problem. The options provided do not include the correct answer. Let's re-evaluate the problem and the options.
We know that 3 − 3 = 3 3 1 = 27 1 . We are looking for an equivalent expression among the given choices. The correct answer should be 3 3 1 . However, this is not among the options. Let's examine the options again:
3 2 1 = 9 1 3 5 1 = 243 1 9 3 1 = 729 1 3 5 1 = 243 1
Final Answer None of the given options are equivalent to 3 − 3 = 27 1 . There seems to be an error in the question or the provided options. However, if we were to choose the closest one, we would need to compare the decimal values. But since none are equal, we cannot choose any of them.
Examples
Understanding exponents and their negative counterparts is crucial in many fields, such as calculating the decay rate of radioactive materials. For instance, if a substance decays at a rate proportional to 2 − t , where t is time, knowing how to manipulate exponents helps predict the remaining amount of the substance after a certain period. This is also applicable in finance when calculating depreciation or compound interest.
