Powers or Indices

# Indices with rational exponents

Indices with Rational Exponents

We have already learned what the concept, indices, is, where n is a natural number.

Fortunately, the same rules apply to indices with rational exponents as well as indices with integer exponents.

Let ; and From all fractions that represent a rational number r, choose the one that has the abbreviated numerator and denominator.

Let's say this fraction is Then:

Example

The example is available to registered users free of charge.

The powers thus defined are calculated in the same way as the powers of whole exponents.

## Rules for calculating powers with rational exponent

A summary of the rules for calculating powers with a rational exponent (the rules are explained in more detail below) is summarized in the table.

Let and Then apply:

 Same base Same exponent Multiplication Division Exponents

### Multiplication and division of Indices with the same bases

Indices with the same bases are multiplied by picking a base and adding the power exponents:

We show this statement by using a different notation of power (with roots) for both powers. Let them be:

Then the following applies:

 We substitute for m and n: We write in root form: We convert to common root: Under the root, we use the rule for multiplying powers of the same bases by adding the exponents: We rewrite in power form: In the exponent, we split a fraction (write the fraction as the sum of two fractions): We simplify the exponent as much as possible: Instead of insert and instead of insert :

When dividing powers of the same bases, we act similarly as when multiplying:

Indices with the same bases are divided by subtracting the exponents:

### Multiplication and division of indices with equal exponents

The rules for multiplying and dividing powers by the same exponents are the same as for powers with natural exponents.

Indices with the same exponents and different bases are multiplied by multiplying the bases and taking the common exponent:

Indices with the same exponents and different bases are divided by dividing the bases and taking the common exponent:

### Exponents

We handle indices with powers by multiplying the power exponents while still retaining the base.

Exponential rule:

material editor: ISMAIL SHOBOLA