Gradual Understanding Mathematics





This chapter inculcates the Student, the ability to

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


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.


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.



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,


  1. =
In algebra when we have

Collect like terms

Product Law of Indices

Note that,

Test yourself

Quotient Law

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


In Algebra

Power law





Test yourself

  1. (

Zero Power Law


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,


Test yourself

Fractional power law

This is where the index/power/exponent is fractional

Line of division

 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.


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;


Similarly, taking


In general,


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


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


How far, how the thing dey go?

General exercise

Simplify the following


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