Algebra
The Basics
An index is a power. We use indices to save space and time.
3 x 3 x 3 x 3 can be written 34. The index is 4. It tells us that 4 threes have been multiplied together.
As you can see, it's a much quicker way of writing the expression. Indices save a lot of time, especially with big numbers:
320 is much easier than 3x3x3x3x3x3x3x3x3x3x3x3x3x3x3x3x3x3x3x3.
And as for 31000. Well, you get the point.
Consequences
Indices are handy.
But new questions start to appear when we use them:
- How do we use them in other calculations?
- Can they be negative?
- Do they have to be whole numbers?
- What does an index of zero mean?
Calculations
Now that we have a great, compact way of writing long products we can start to combine our expressions.
Think about:
- 52 x 53
- 75 / 74
- 34 + 42
- (42)3
52 x 53 = 5x5 x 5x5x5 = 55
The indices were just added. We're just counting up how many fives have been multiplied together. Two fives and another three makes five fives multiplied together.
75 / 74 = (7x7x7x7x7) / (7x7x7x7) = 71 = 7
The second index was subracted from the first. The four sevens multiplied on the bottom cancelled four of the sevens on the top leaving one seven behind.
34 + 42 = 3x3x3x3 + 4x4 = 81 + 16 = 97
Here, we don't have lots of the same number being multiplied. We can't use the indices to make things easier so we write the expressions as ordinary numbers and add them.
(42)3 = (4x4)3 = (4x4) x (4x4) x (4x4) = 46
The indices were multiplied. We had three lots of two fours multiplied together. That's six fours altogether.
Negative Indices
Just by using the rules above we start to come across more of our questions:
32 / 34 = 32-4 = 3-2
But what does that mean? Let's write the calculation without indices:
(3x3) / (3x3x3x3) = 1 / (3x3). The two threes on top cancel two of the threes on the bottom.
But we can write 1 / (3x3) as 1 / 32
So negative indices give us one over our number:
- 3-2 = 1 / 32
- 5-4 = 1 / 54
- (-3)-5 = 1 / (-3)5
An Index of Zero
When you divide a number by itself you get one.
- 5 / 5 = 1
- 123 / 123 = 1
- 0.264 / 0.264 = 1
- (-3/5) / (-3/5) = 1
So 35 / 35 = 1.
But 35 / 35 = 35-5 = 30
30 = 1
In fact, any number with an index of zero is one.
Fractional Indices
91 = 90.5 x 90.5
But 9 = 3 x 3. So, 90.5 = 3.
90.5 is the square root of 9 because when you multiply it by itself you get 91
In a similar way, other fractions can be seen as roots:
81/3 is the cube root of 8 because 81/3 x 81/3 x 81/3 = 81. So 81/3 = 2.