1.6 I) Fractional Indices – Part 2
A Bit of Proof
With indices, it is important that you fully understand all of the rules. I am going to show you an example that we can answer by using a few different rules to obtain the same answer.
Example 1
What is:
The three different methods that I am going to show you are:
1) Multiplication
2) Using fraction powers
3) Raising a power to another power
1) Multiplication
When we are multiplying powers that have the same base, we add the powers together. The base of both of the terms above is 16, which means that we can add the powers.
Anything to the power of 1 is itself, thus meaning that 161/2 x 161/2 is 16.
2) Using Fraction Powers
A fractional power of a half means that we need to square root.
The square root of 16 is 4.
The method gives us the same answer as before.
3) Raising a Power to Another Power
The question is multiplying 16 ½ by itself twice. This means that we are squaring 16 ½. Therefore, we can write the calculation as:
We are now raising a power to another power. When we are raising a power to another power, we multiply the two powers.
We can write 2 as 2/1 and when we multiply fractions, we multiply straight across.
Anything to the power of 1 is the base.
The answer is 16.
You can see from this question that all of the different rules give the same answer.
Example 2
What is:
Whenever you are given a negative power, it is best to deal with this negative power first. This is because it is very easy to forget to deal with the negative power later on. A negative power means that we need to reciprocate our base and change the power into a positive one. Reciprocating the base means dividing 1 by the base. The original base is 9 and the reciprocal of 9 is 1/9.
The base for our question is a fraction and whenever we have a fraction, it is best to split it up into the numerator to the power and the denominator to the power.
The numerator is 1 and we know that 1 to the power of anything is 1. Therefore, the numerator for our answer is going to be 1.
The denominator for our base is 9 and it has the power 3/2.
The denominator of the power is 2, which tells us that we need to square root 9, which is 3. The numerator of the power tells us what we raise the root to. The numerator of the power 3/2 is 3, which tells us that we need to cube 3.
The denominator of our answer will be 27.
The final step is to combine the numerator and denominator to obtain the final answer of 1/27.