Now we are going to study the **properties of the powers**.

What are the power properties for? Because they allow us to operate with the powers and thus be able to simplify much more complex expressions.

We’re going to start by defining what a power is.

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## What is a power

What is a power?

A power is a product of factors that are repeated a certain number of times. The repeating factor is the base and the number of times it is repeated is the exponent.

In this example, 3 is multiplied 4 times, so it reads that 3 is raised to 4.

In general it can be represented as:

Where a is the base and n is the exponent.

## Powers with negative base

A particular case of powers is when the base is negative. In this case, the result will depend on whether the exponent is even or odd. In general:

We must be very careful if the sign – is not within the parenthesis, since in that case, it is not elevated with the power, but goes to part:

That is, the sign – is completely independent of power. We have a minus sign followed by a two squared.

We continue with the properties of the powers.

## Properties of Powers

### Powers with exponent one

Any value raised to 1 results in the same value:

Example:

### Multiplication of powers

Multiplication of powers with the same base: The base is maintained and the exponents are added: Example:

### Division of power

Division of powers with the same base: The base is maintained and exponents are subtracted:Example:

### Other power

High power to another power: The base is maintained and the exponents are multiplied:

You can find the power without parentheses, but that is not correct. It should be in parentheses to indicate that all power is being raised to another power.

Example:

### Multiplication elevated to power

Multiplication elevated to a power: This is the same as each value elevated to the same power:

### Quotient elevated to a power

Quotient elevated to a power: Is equal to the numerator and denominator elevated to the same power:

Example:

If we can operate within the parenthesis, we can also solve the power without applying this property, only resolving the parenthesis and then elevating it to the exponent:

### Powers with negative exponent

Negative exponent powers: A value elevated to a negative power is equivalent to 1 divided by the value elevated to the same positive power:

Example:

As a particular case of this property is the inverse of a number, which is any value raised to -1:

High fraction to negative exponent: In this case the fraction is turned over and the exponent becomes positive:

### Rational exponent power

Example:

The denominator of the exponent becomes the index of the root

### Rational and negative exponent power

Example: