Which expressions are equivalent to $\frac{1}{81}$? Check all that apply.
$8^1$
$9^2$
$9^t$
$9^t$
$9^{\frac{1}{2}}$
$9^{16}$
$0^6, 0^3$
$0^5, 0^7$
Answer (1)
Express 81 1 as 9 − 2 .
Evaluate each given expression.
Identify that 9 t is equivalent to 81 1 only when t = − 2 .
The expressions equivalent to 81 1 are 9 t when t = − 2 .
Explanation
Understanding the Problem We are given the value 81 1 and a list of expressions. Our goal is to identify which expressions are equivalent to 81 1 .
Expressing 1/81 as a power of 9 First, let's express 81 1 as a power of 9. Since 81 = 9 2 , we have 81 1 = 9 2 1 = 9 − 2 .
Evaluating Each Expression Now, let's examine each expression to see if it is equal to 9 − 2 = 81 1 .
8 1 = 8 . This is not equal to 81 1 .
9 2 = 81 . This is not equal to 81 1 .
9 t . This expression is equal to 81 1 only if t = − 2 .
9 t . This expression is equal to 81 1 only if t = − 2 .
9 2 1 = 9 = 3 . This is not equal to 81 1 .
9 16 . This is a very large number and not equal to 81 1 .
0 6 = 0 and 0 3 = 0 . These are not equal to 81 1 .
0 5 = 0 and 0 7 = 0 . These are not equal to 81 1 .
Identifying Equivalent Expressions From the above analysis, only 9 t is equivalent to 81 1 when t = − 2 .
Final Answer Therefore, the expressions equivalent to 81 1 are 9 t when t = − 2 .
Examples
Understanding powers and exponents is crucial in many fields, such as finance and computer science. For instance, calculating compound interest involves exponential growth. If you invest $100 at an annual interest rate of 5%, the amount you have after t years is given by 100 ( 1 + 0.05 ) t . Similarly, in computer science, the efficiency of algorithms is often expressed using exponential notation, such as O(2^n) for an algorithm that doubles its runtime with each additional input.
