Indices
3³ ('3 cubed' or '3 to the power of 3') and 5² ('5 squared' or 5 'to the power' of 2) are example of numbers in index form.
3³ = 3×3×3
2¹ = 2
2² = 2×2
2³ = 2×2×2
etc.
The ² and ³ are known as indices. Indices are useful (for example they allow us to represent numbers in standard form) and have a number of properties.
Laws of Indices
There are several rules for dividing and multiplying numbers written in index form. These properties only hold, however, when the same number is being raised to a certain power. For example, we cannot easily work out what 2³×5² is, whereas we can simplify 3²×3³ .
Multiplication
When we multiply together index numbers, we add the powers. So:
ya × yb = ya+b
Examples
x2 × x3 = x5
54 × 5-2 = 52 (because 4 + (-2) = 2)
But there is no easy way of calculating 54 × 33 because 5 and 3 aren't the same number!
Division
When dividing index numbers, we subtract the power of the number we are dividing by from the power of the number being divided. So:
ya ÷ yb = ya - b
Examples
x2/x3 = x-1
72 ÷ 7-5 = 77
Brackets
(ya)b = ya×b
Examples
(x2)3 = x6
(53)2 = 56
Further Index Properties
Anything to the power 0 is equal to 1. So 30 = 1, -8240 = 1 and x0 = 1.
Negative Indices
If you have a number raised to a negative power, this is equal to 1 divided by the number raised to the power made positive. In other words:
n-a = 1/na.
Examples
n-1 = 1/n.
3-2 = 1/32 = 1/9
(½)-3 = 23 = 8
Fractional Indices
A fractional power means that you have to take a root of the number. For example, 4½ means take the square root of 4 = 2. Similarly, x1/3 means take the cube root of x.
We can use the rule (ya)b = ya×b to simplify complicated index expressions.
Example
(1/8)-1/3; = [(1/8)-1]1/3 = [8]1/3 = 2
Inverse
The inverse of something has the opposite effect of that thing. Suppose you multiply something by 2. Clearly the "opposite effect" is to divide by 2.
Similarly, if you raise a number x by a power b, the inverse of this would be to raise it by the power of 1/b. This is because (xb)1/b = x1. So if we raise to the power of b and then to the power of 1/b, we end up where we started. So raising to the power of 1/b must 'undo' what we did by raising to the power of b.
For example, the inverse of cubing something is to take the cube root. If we do 23, we get 8. If we then cube root this, we get 81/3 = 2.
Reciprocals
The "reciprocal" of something means 1 over that something. So the reciprocal of y is 1/y = y-1 . The important thing about reciprocals is that if you multiply a number together with its recipricol, you get 1. So 1/y × y = 1. The reciprocal of 1/2 is 2 because ½ × 2 = 1.
Every number has a reciprocal except zero. Zero doesn't have a reciprocal because you are not allowed to divide by zero, so we can't work out 1/0.
y-1 is sometimes pronounced "y inverse", because multiplying by 1/y is the inverse (opposite) of multiplying by y.
