An index (plural: indices) is the power, or exponent, of a number. For example, has an index of 3.
A surd is an irrational number that can be expressed with roots, such as or .
The manipulation of indices and surds can be a powerful tool for evaluating and simplifying expressions.
Let's start with some basic rules for operations with indices:
Surds are just numbers with fractional indices, e.g. . Any operation with indices can be applied to surds, and indices and surds are related through this rule:
This allows us to group numbers together into forms that can be more convenient. Here are a couple examples:
Sometimes surds will appear in the dominator of an expression. You can rationalize the denominator by applying the following technique to a fraction of the form :
Application and Extensions
If you write the prime factorization of, what is the sum of indices of the factors?
If you recognize that , the answer falls quickly into place:
So, the sum of the indices is simply 8.
If and , what is ?
Multiply the two expressions together to get the 's to cancel out: