Which expression represents a^ (m/n) as a radical?

• A. The n-th root of a^m
• B. a^(m/n)
• C. The m-th root of a^n
• D. The n-th root of a^m raised to the n

Answer: (A)

Converting a fractional exponent to a radical uses the idea that taking the nth root is the same as raising to the 1/n power. So a^(m/n) equals (a^m)^(1/n), which is the nth root of a^m. This matches the radical form: the n-th root of a^m.

For example, with a = 4, m = 3, n = 2, a^(m/n) = 4^(3/2) = (4^3)^(1/2) = sqrt(64) = 8, which is exactly the square root of 4^3.

The other expressions don’t give that same value. The m-th root of a^n equals a^(n/m), not a^(m/n). And taking the n-th root of a^m raised to the n, i.e., (√[n]{a^m})^n, simplifies to a^m, not a^(m/n).