Gradual Understanding Mathematics |

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**INDICES**

__Objectives__

This chapter inculcates the Student, the ability to

- Recognize and express numbers in their different form.
- Apply the four basic operation of addition, subtraction, multiplication and division

**INTRODUCTION**

You are welcome to the world of indices; you are probably familiar with this expressions like and { raised to the power of 2 or 3 respectively}. Note that 2 and 3 tell you the number of times you multiply .

In general, where is multiplied by itself m times.

Base

Infact, this idea could include all kinds of numbers as indices, not just positive integers. We can for example, have and so on, and all these have their own particular meaning.

There are only a few rules of indices and they appear quite simple. This is deceptive for many Students, a tricky topic can be of trouble. At the same time it is extremely important for later work. So it really a while on this section, making sure that you completely understand the rules and are confident that you can answer the corresponding examples. Lets begin from simple to complex.

__Laws of indices__

__Product law__

This means a total of multiplied altogether. Under the condition that the base are the same.

Again,

__Analysis __

If the base are the same, take one as the base and add whatever index/power/exponent given.

Note that, if a letter or base has no power, it is one in power.

For instance,

__Examples__

- =

In algebra when we have |

Collect like terms

Product Law of Indices |

POWER LAW |

Note that,

__Test yourself__

__Quotient Law__

If given the same base, take one and subtract the power.

** Example**:

In Algebra |

__Power law__

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__Examples __

__Test yourself__

- (

__Zero Power Law__

__Examples __

__Test yourself__

__Negative power law__

The negative sign at the power, is fractional which gives a fractional outcome to the base and power but, the result excludes the negative sign.

In particular,

__Example__

__Test yourself__

__Fractional power law__

This is where the index/power/exponent is fractional

Numerator |

Line of division |

Denominator |

** Back to primary school**

For the fractional power law, the denominator of the power becomes the root and numerator becomes the index/power or exponents.

Therefore,

We can further extend the value that the indices m and n can take, if we allow them to be fractions and see what this leads to.

We can use power law

So, when we square , we end up with .

We already have a term for this, the square-root of written;

So,

Similarly, taking

So,

**In general, **

__Examples __

- If we need evaluate for example, we would probably use the second alternative

Instead,

Since you might know that the fourth-root of 16 is a big number which you then have to square root

Otherwise,

How far, how the thing dey go? |

General exercise

Simplify the following