The laws of indices

The laws of indices

In the following laws a is the common base, m and n are the indices. Each law has an example of its use alongside.

We need to study these laws carefully in order to understand the significance of each.

Law 1, you have already met, it enables us to multiple numbers gi in index form that have a common base. In the example the common base is 2, the first number raises this base (factor) to the power 2 the second raises the same base to the power 4. In order to find the result we simply add the indices.

Law 2, we have again used when dividing numbers with a common '"' base, in this case the base is 3. Note that since division is the opposite arithmetic operation to multiplication. It follows that we should perform the opposite arithmetic operation on the indices, that of subtraction. Remember we always subtract the index in the^-" denominator from the index in the numerator.

Law 3 is concerned with raising the powers of numbers. Do not m^, this law up with law 1. When raising powers of numbers in index form, we multiple the indices.

Law 4, you have also met, this law simply states that any number raised to the power 0 is always 1. Knowing that any number divi by itself is also 1, we can use this fact to show that a number rais to the power 0 is also 1. What we need to do is use the second la concerning the division of numbers in index form.

We know that ⁹/₉ = 1 or 3²/3² = 3^2-2 = 3⁰ = 1 which shows that 3⁰ =1 and in fact because we have used the second law of indices, this must be true in all cases.

Law 5, this rather complicated looking law, simply enables us to the decimal equivalent of a number in index form, where the index is a fraction. All that you need to remember is that the index nun above the fraction line is raised to that power and the index num below the fraction line has that number root.

So, for the number 8, we raise 8 to the power 2 and then take the cube root of the result. It does not matter in which order we perform these operations.          So, we could have just as easily taken the cube root of 8 and' then raised it to the power 2.

           

Law 6 is very useful when you wish to convert the division of a number to multiplication. In other words bring a number from underneath the division line to the top of the division line. As the number crosses the line we change the sign of its index. This is illustrated in the example that accompanies this law.

The following examples further illustrate the use of the above laws, when evaluating or simplifying expressions that involve numbers and symbols.