The standard form is like the short form of a particular function. There are  different standard forms, like the standard form of an equation, standard form of a polynomial and the standard linear equation. The standard form formula is  used to help in finding the general representation for the different types of notations. 

The exponential function is that function that is represented in the form x^{m} where x is base and m is the exponent. The exponential curve depends on the exponential function, and it also depends on the value of m in the expression x^{m}. 

The index is the power or exponent that is raised to a number or a variable. For example, in number 2^4, 4 is the index of 2. The plural form of index is indices.

Indices are a convenient tool in mathematics to compactly denote taking power or roots of a number. Taking power is simply a case of repeated multiplication of a number. Whereas taking root is equivalent to a fractional power of the number.