**Introduction****Indices are a useful way of more simply expressing large numbers. They also present us with many useful properties for manipulating them using what are called the Law of Indices.**

**What are Indices?****The expression 2 ^{5} is defined as follows:**

**We call "2" the base and "5" the index.**

**Law of Indices****To manipulate expressions, we can consider using the Law of Indices. These laws only apply to expressions with the same base, for example, 3^{4} and 3^{2} can be manipulated using the Law of Indices, but we cannot use the Law of Indices to manipulate the expressions 3^{5} and 5^{7} as their base differs (their bases are 3 and 5, respectively).**

**Six rules of the Law of Indices****Rule 1:**

**Any number, except 0, whose index is 0 is always equal to 1, regardless of the value of the base.**

**An Example:**

**Simplify 2 ^{0}:**

**Rule 2:**

**An Example:**

**Simplify 2 ^{-2}:**

**Rule 3:**

**To multiply expressions with the same base, copy the base and add the indices.**

**An Example:**

**Simplify : (note: 5 = 5^{1})**

**Rule 4:**

**To divide expressions with the same base, copy the base and subtract the indices.**

**An Example:**

**Simplify**

**:**

**Rule 5:**

**To raise an expression to the nth index, copy the base and multiply the indices.**

**An Example:**

**Simplify (y ^{2})^{6}:**

**Rule 6:**

**An Example:**

**Simplify 125 ^{2/3}:**

**You have now learnt the important rules of the Law of Indices and are ready to try out some examples!**

**Go to the next page for the first of many questions and fully worked out solutions for you to practice.**

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That's what Carl Linnaeus, the father of modern taxonomy, did for his book *Systema Naturae*. Published in 13 editions from 1735 to 1770, *Systema Naturae *classified and named organisms and minerals. It gave scientists an understanding of which species were alike based on the number of shared categories.

Before publishing his taxonomies, Linnaeus had to organize everything himself. But instead of writing his classifications in a book that could easily run out of space, he put each organism and mineral on its own piece of paper. That way, Linnaeus could have a file of everything he recorded. He could easily retrieve data on any organism and mineral and reposition any of them that may have been placed in the wrong spot. Most importantly, new discoveries could always be added as he published new editions of *Systema Naturae*.

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The Swedish naturalist and physician Carl Linnaeus (1707-1778) ? the father of modern taxonomy - has been described as a ?pioneer of information retrieval?. But exactly how he was able to master such vast amounts of data has remained something of a mystery.

Staffan Mueller-Wille from the Centre for Medical History at the University of Exeter in the UK recently received a major grant from the Wellcome Trust to get to the bottom of Linnaeus? method of data processing.

Speaking at the annual meeting of the British Society for the History of Science in Leicester, UK?on Saturday 4?July, Mueller-Wille will reveal his preliminary findings of research on Linnaeus? manuscripts held June 16 at the Linnaean Society of London.

Linnaeus had to manage a conflict between the need to bring information into a fixed order for purposes of later retrieval, and the need to permanently integrate new information into that order, says Mueller-Wille. ?His solution to this dilemma was to keep information on particular subjects on separate sheets, which could be complemented and reshuffled,? he says.

Towards the end of his career, in the mid-1760s, Linnaeus took this further, inventing a paper tool that has since become very common: index cards. While stored in some fixed, conventional order, often alphabetically, index cards could be retrieved and shuffled around at will to update and compare information at any time.

?Although a seemingly mundane and simple innovation, Linnaeus' use of index cards marks a major shift in how eighteenth-century naturalists thought about the order of nature,? says Mueller-Wille. The natural world was no longer ordered on a fixed, linear scale, but came to be seen as a map-like natural system of multiple affinities.

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