We square numbers to calculate areas. We cube numbers to calculate volumes. In some cases, we may need to multiply a number by itself more than three times. Think of this as an extension to the skill of multiplying ... super multiplying, if you will.
Not only may the answers become difficult to calculate, but even writing the product out could become very tedious.
Interestingly, the notation that we use was only adopted quite recently (historically speaking). Prior to the 17th century a variety of notations were proposed, many of them involving words or abbreviations of words. By 1631, the English astronomer, mathematician and generally curious Thomas Harriot (1560-1621) had adopted the notation a, aa, aaa, aaaa, aaaaa to denote the products. This made quite good sense for small numbers, but most of us would find it tedious writing 20 or 30 'a's in this way. [By the way, Thomas Harriot was the first person to build a telescope and use it to examine the heavens ... not Galileo.]
Just three years later (in 1634), French mathematician Pierre Hérigone (or Herigonus) (1580-1643) wrote a, a2, a3, etc., in Cursus Mathematicus, which he published in five volumes from 1634 to 1637. [As an aside, it was Pierre who created the perpendicularity sign that we use today, ⊥.]
Also living in France, the Scottish mathematician, James Hume (fl.1639) created the first true exponential notation in 1636. You can see that the 1630s was a busy and creative decade! What James Hume did, in writing about the algebra of Vieta, was to write the base and elevate the exponent as a superscript to the right of the base (where we write our indices today). The difference was that he used Roman numerals in lower case, e.g. 4ᵛ meant four to the power five.
The following year, René Descartes (1596-1650) used our modern notation for the first time, in his La Géométrie (Geometry). It should be noted that, although he used terminology that we would instantly recognise, such as a³, he usually did not use 2 as an exponent, preferring to write aa instead of a², and he only used positive numbers as powers.
One of the great advantages of this notation is that, by writing the exponent on a different level (as a superscript), it could be clearly identified and treated differently from the base numeral or pronumeral. Another related advantage is that the notation does not allow most students to become confused between 2³ and 3². In other words, the assymmetry in the notation reflects the fact that powers are not commutative. It may be true that 2 + 3 = 3 + 2 and 2 × 3 = 3 × 2 ... but 2³ ≠ 3².
In 1659, only a few years later, the English mathematician, John Wallis (1616-1703), gave meaning to negative and fractional indices, and his contemporary, Sir Isaac Newton (1642-1727) appears to have been the first to consider pronumerals as powers (e.g. aⁿ) in private letters to Gottfried Wilhelm Leibniz.