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Powers, Indices, Roots, Radicals and Surds

An illustration that two multiplied by itself six times equals two to the power sixPowers and indices were created as mathematicians developed algebra and needed a new notation to describe some of the numbers and concepts that they were encountering.  They were also needing to calculate large numbers for the sciences.

There is also another matter for you to consider.  I am sure that you have already mastered the four basic operations on numbers … addition, subtraction, multiplication and division.  Have you ever considered that someone must have been the first to ask, “What if we keep on adding?”?

“What if we keep on adding four, let’s say twenty times … what will we get?”  You can see that writing 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 would be tedious.  Multiplication was invented/developed to resolve that problem.  We would now simply write 20 × 4 meaning “twenty lots of four” and, if we know our tables, rapidly conclude that the result is eighty (80).

Subtraction responds in a similar way.  We could replace – 3 – 3 – 3 – 3 – 3 – 3 – 3 – 3 – 3 – 3 – 3 – 3 – 3 – 3 – 3 – 3 – 3 – 3 – 3 – 3 with 20 × (-3) = -60.

Mathematicians simply asked, “What if we multiplied the same number many times?” and 5 × 5 × 5 × 5 × 5 × 5 × 5 × 5 × 5 × 5 × 5 × 5 × 5 × 5 × 5 × 5 × 5 × 5 × 5 × 5 × 5 = 5²¹ is the result!

The entire expression 5²¹ is called a power and mathematicians speak of “working with powers.”

We call 5 the base number or, simply, the base.  Curiously, the superscript has four different names, depending upon the branch of mathematics you are studying, or what you are particularly interested in.  It is usually called the index and we speak of index laws when we discuss how to combine them under certain operations.  It is called the exponent and many functions are called exponential functions when they have a pronumeral (like x) in this position.  The superscript is also called a/the power, and we speak of “five to the power twenty-one.”  And there is another branch of mathematics that concentrates on the behaviour of these numbers, and calls them logarithms … but a different notation is used.  We will address that when the time comes.

Roots, Radicals and Surds

History of the Notation

A power, identifying its base and index and showing that the index can also be called an exponent, a power, or a logarithmWe 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.

How to Learn Your Cubic Numbers

In my experience, top high school mathematics students benefit enormously from knowing the cubic numbers up to 12 cubed. In this video I do three things:

  • introduce you to each of the cubes,
  • show you features to help you remember them, and then
  • introduce you to a 'game' so you will learn most of them rapidly.

I have used this method with great success even with students much younger than high school age. I am confident that, if you will 'play the game,' you will know your cubes quite thoroughly as well!

A Quick Method for Cubing Numbers from 10 to 99

If I needed to cube a two digit number I would normally reach for my calculator ... even knowing what I am about to show you. However, I am sharing this skill with you (and showing you the theory behind it) in order to

  • demonstrate a link between the theory of algebra and numbers,
  • increase your understanding of number manipulation,
  • help you see place notation used in a different way,
  • demonstrate a use of Pascal's Triangle, and
  • reinforce your understanding of cubic expansions.

Having said that, I still occasionally cube numbers in this way simply because I can, it is 'fun,' and it keeps my mind active. I commend it to you and suggest you practise it until you can use it with finesse! This is an opportunity to understand your algebra better, understand place notation more thoroughly, and practise using your knowledge of cubes up to nine cubed.

I hope you enjoy it.

I don’t know if you remember me, but you really helped me last year and I just wanted to tell you I came first in my maths half yearly with 100%. A lot of it goes to you. Thanks for your help.
Jashan M (student, 2013)

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