# Nix the Tricks: Fractional indices

This post on fractional indices is the second in a series featuring ideas from Tina Cardone’s book ‘Nix the Tricks’.

A recent class had difficulty in moving from integer to fractional indices. I used some of the ideas from ‘Nix the Tricks’ and began by clarifying language.

A typical way of describing 2^{3} is ‘Two multiplied by itself three times’ which could mean either

(2 x 2) x (2 x 2) x (2 x 2)

or ((2 x 2) x 2) x 2)

Given the lack of clarity students will sometimes use 2 x 3 as a fall back.

As an alternative indices can be described as indicating the number of factors of the base. So 2^{3} is ‘three factors, each of which is two’.

A further advantage of this description of exponents is that it can be expanded to fractional indices and in doing so lessens the jump from integer indices.

Hence 8^{1/3} is read as ‘one third of a factorisation of 8’. To find one third of something you must break the whole into three parts. A factorisation of 8 is

8 = 2 x 2 x 2

One third of the factorisation gives the answer 2.

Representing 8^{2/3 } involves keeping two thirds of the factorisation giving the answer 2 x 2 = 4

This method struck students as being a natural progression between whole number and fractional indices. Linked to this post is a resource that develops this representation. Give it a try and reply to this post sharing how students get on.

We have recently added the __‘Nix the Tricks’__ book both in its entirety and divided into chapters. The discussion in this post arises from __Chapter 5__ of the book.

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