Which expression represents a^ (p/q) as a radical?

Which expression represents a^ (p/q) as a radical?

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

Answer: (B)

Rational exponents can be read as radicals: a^(p/q) is the q-th root of a^p. This follows from the rule a^(m/n) = (a^m)^(1/n), so a^(p/q) = (a^p)^(1/q) = the q-th root of a^p. This form is the exact radical representation because taking the q-th root and then raising to the q restores a^p, matching the original exponent p/q. The other ways don’t match a^(p/q) in general. The p-th root of a^q equals a^(q/p), which is different from a^(p/q) unless p = q. The expression that involves the p-th root of a^q raised to the q simplifies to a^{q^2/p}, which again is not the same as a^(p/q).

Rational exponents can be read as radicals: a^(p/q) is the q-th root of a^p. This follows from the rule a^(m/n) = (a^m)^(1/n), so a^(p/q) = (a^p)^(1/q) = the q-th root of a^p. This form is the exact radical representation because taking the q-th root and then raising to the q restores a^p, matching the original exponent p/q.

The other ways don’t match a^(p/q) in general. The p-th root of a^q equals a^(q/p), which is different from a^(p/q) unless p = q. The expression that involves the p-th root of a^q raised to the q simplifies to a^{q^2/p}, which again is not the same as a^(p/q).