To solve the given expression \(\left(15^{1/\pi}\right)^5\), we follow a step-by-step process.
First, interpret the notation correctly:
[tex]\[
\left(15^{1/\pi}\right)^5
\][/tex]
Next, remember the property of exponents which states: \((a^m)^n = a^{m \cdot n}\). We can use this property to simplify our expression. In our case, \(a = 15\), \(m = 1/\pi\), and \(n = 5\):
[tex]\[
\left(15^{1/\pi}\right)^5 = 15^{(1/\pi) \cdot 5}
\][/tex]
Now we need the value of \((1/\pi) \cdot 5\):
[tex]\[
(1/\pi) \cdot 5 = \frac{5}{\pi}
\][/tex]
Thus, our expression is simplified to:
[tex]\[
15^{5/\pi}
\][/tex]
Now let's evaluate \(15^{5/\pi}\). Given the true results, we know that:
[tex]\[
15^{1/\pi} \approx 2.3678897349212167
\][/tex]
Raising this value to the power of 5 gives us:
[tex]\[
(2.3678897349212167)^5 \approx 74.44017305013445
\][/tex]
Hence, the value of the original expression \(\left(15^{1/\pi}\right)^5\) is approximately \(74.44\).
Since none of the provided options match this result exactly:
- A. 1
- B. 8
- C. 15
- D. 243
None of the given answer choices is correct.
