Final answer:
The nth root of a number b can be rewritten in exponential form as b raised to the power of 1/n. This removes the radical sign and converts it into a fractional exponent. For more complex numbers, exponential functions can be used by applying logarithmic identities and rules for combining exponents.
Step-by-step explanation:
To express the nth root of a number b in exponential form, you can rewrite it using a fractional exponent. In this representation, the radical sign is removed, and the expression is given as b raised to the power of 1/n. Therefore, the nth root of b, which is √b, becomes b1/n.
For example, the square root of x, which is √x or x1/2, represents x raised to the power of 0.5. To deal with exponential operations such as multiplication, you can add the exponents when the bases are the same, as in 51 · 51 = 52, and to deal with division, you subtract the exponents.
Using this knowledge, you can also express a base b raised to an arbitrary number n as an exponential function using natural logs (ln) and e, as shown in the equation bn = en ln b, where e is approximately equal to 2.7183.
To combine the nth root and exponential terms, you can manipulate exponential expressions. For example, to express 3 to the power of 1.7, you can take the tenth-root of 3 and raise it to the 17th power: 31.7 = (31/10)17 = 317/10.