Which is equivalent to $243^{\frac{1}{5}}$?
Answer (1)
We are asked to find the value of 24 3 5 1 .
We need to find a number that, when raised to the power of 5, equals 243.
We calculate 3 5 = 3 × 3 × 3 × 3 × 3 = 243 .
Therefore, 24 3 5 1 = 3 .
Explanation
Understanding the Problem We are asked to find the value of 24 3 5 1 . This is equivalent to finding the fifth root of 243. In other words, we need to find a number that, when raised to the power of 5, equals 243.
Checking the Options We need to check which of the given options, 3, 6, 9, or 12, when raised to the power of 5, equals 243.
Calculating 3 to the power of 5 Let's calculate 3 5 . This means 3 × 3 × 3 × 3 × 3 . We can calculate this as follows: 3 × 3 = 9 9 × 3 = 27 27 × 3 = 81 81 × 3 = 243 So, 3 5 = 243 .
Finding the Fifth Root Since 3 5 = 243 , the fifth root of 243 is 3. Therefore, 24 3 5 1 = 3 .
Examples
Understanding exponents and roots is crucial in many fields, such as finance. For example, if you want to calculate the annual interest rate needed to grow an investment of $1000 to 1464.10 in 5 ye a rs , yo u w o u l d u se t h e f or m u l a : ann u a l r a t e = (\frac{future value}{present value})^{\frac{1}{number of years}} - 1 . I n t hi sc a se , i tw o u l d b e (\frac{1464.10}{1000})^{\frac{1}{5}} - 1 = (1.4641)^{\frac{1}{5}} - 1 = 1.08 - 1 = 0.08$, or 8%.
