Question

Which of the following choices rightly explains what a rational exponent, such as

3
4
4
3
​
, represents?

A) The denominator 4 of the rational exponent is the exponent that represents the power of the base, and the numerator 3 is the index of a root.

B) None of these choices explains it.

C) The denominator 3 of the rational exponent is the index of a root, and the numerator 4 of the rational exponent represents the power of the base.

D) The denominator 4 of the rational exponent is the index of a root, and the numerator 3 of the rational exponent represents the power of the base.

E) The denominator 4 of the rational exponent is the index of a root, and the numerator 3 of the rational exponent represents the power of the base.

F) It is a fraction that is the exponent of a number.

G) The denominator 4 and the numerator 3 are the exponent of the rational exponent.

Answer 1

The correct explanation of a rational exponent like 3/4 is that the denominator (4) is the index of a root and the numerator (3) signifies the power of the base. The rational exponent signifies taking the fourth root of the base cubed.

Step-by-step explanation:

The rational exponent such as 3/4 represents the combination of a root and a power operation on a base number. The correct explanation for what a rational exponent represents is: the denominator of the rational exponent is the index of a root, and the numerator of the rational exponent represents the power of the base. Therefore, the correct choice is:

D) The denominator 4 of the rational exponent is the index of a root, and the numerator 3 of the rational exponent represents the power of the base.

By this definition, for a base number x, an exponent such as 3/4 is interpreted as x^(3/4) = (x^3)^(1/4) or the fourth root of x cubed, that is, √x³. This shows the connection between exponents and roots and how they work together to simplify calculations involving rational exponents.