Final answer:
The equivalent expression for the 5th root of the square root of 1,215 raised to the power of x is 5^(x/10), which translates to the same as taking the square root of 5 raised to the power of x/2.
Step-by-step explanation:
The 5th root of (square root of 1,215) to the power of x can be expressed in equivalent form using properties of exponents and roots. The square root is equivalent to raising something to the 1/2 power, so taking the 5th root is the same as raising to the 1/5 power. When you compose these two operations and raise this to a power 'x', it's equivalent to multiplying the exponents.
Let's represent the square root of 1,215 (which is 5 because sqrt(1215) = sqrt(5*5*5*7*7)) as 5^(1/2). Then, we'll take the 5th root, writing it as 5^(1/2 * 1/5). Finally, we raise this to the power of x, which gives us 5^(1/2 * 1/5 * x).
Therefore, combining the exponents (1/2 * 1/5 = 1/10), the expression simplifies to 5^(x/10), which is equivalent to option b) Square root of (5^(x/2)) since taking the square root is the same as raising to the power of 1/2.