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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
p/s: please write the index notation in the correct way. It might confuse you as well as ME.
Untill then,
Thanks for reading this entry.
Love,
Cicer Ziha
