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Exponential notation is a sleek and swift way to portray numbers or expressions that involve repeated multiplication of the same factor. This compact form accentuates the practice of raising a base to a certain power to represent large or small quantities efficiently.

Let’s take the bread-and-butter example from our exercise: 'b cubed'. Rather than writing out 'b' multiplied by itself three times as 'b × b × b', exponential notation allows us to condense this into a much more succinct presentation: using the number '3' to denote how many times 'b' should be multiplied by itself, resulting in the neat expression 'b^3'. This concise method is not only a timesaver but also becomes critical when dealing with more complex mathematics or large numbers. Exponential notation enables clear communication of mathematical ideas and is a cornerstone of algebraic expressions.

The duo that forms the very foundation of exponential notation is the 'base' and 'power'. Understanding these terms is akin to mastering the alphabet before forming words and sentences.

The 'base' is essentially the repeated factor, the value that is being multiplied by itself. In our textbook's example, 'b' assumes that role. The 'power', on the other hand, is the exponent that tells us the number of times the base is to be used in multiplication. The term 'cubed' in normal speech corresponds to the power of 3 in mathematical lingo.

When wielding exponential notation, it's paramount to correctly identify the base and power. A simple rule of thumb is: the base appears as the solid groundwork, the number or expression that's multiplied, while the power sits aloft, compactly encapsulating the count of multiplications in a small, superscripted numeral. This dynamic is what gives exponential expressions their potency and why they're so widely used in scientific, engineering, and financial calculations.

Exponents, the small yet mighty figures perched above the base, are the cornerstones of exponential expressions. These numerical superscripts are not mere ornaments; they embody a directive—dictating how many times the base is to embrace itself in a multiplication dance.

In essence, exponents are shorthand for repeated multiplication. Returning to our 'b cubed' illustration, the exponent is '3'. It's important to realize that exponents can go beyond whole numbers. They can be fractions, negative numbers, or even zero—each carrying its own set of rules on how it modifies the base.

For example, when an exponent is zero, as in 'b^0', the result is always 1, regardless of the base. Conversely, a negative exponent, say 'b^-3', suggests the reciprocal process: one over the base raised to the positive version of our exponent, or 1/(b^3). And if our exponent is a fraction, it signals the extraction of roots, inviting a new layer of mathematical intrigue.

Understanding exponents is not just a mathematical skill—it's a key that unlocks a vast expanse of numerical computation and allows us to fathom numbers on a cosmic scale, as well as miniscule phenomena.