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# Using Properties of Exponents

The properties of exponents are a set of tools that can be used to simplify expressions with exponents.

## Properties of Exponents

The properties of exponents allow expressions with exponents to be rewritten.

## Product of Powers Property

### Rule

When powers with the same base are multiplied, they can be written as one power by adding the exponents.

For example, can be expressed as follows using this rule. This rule can be explained by writing the powers as repeated multiplication.

This rule is valid for all values of and when For the rule to be true, when it is necessary that and that Note that the rule might lead to non-real solutions when and at least one of the exponents is not an integer.

## Quotient of Powers Property

### Rule

When powers with the same base are divided, they can be written as one power where the exponent in the denominator is subtracted from numerator's exponent. For example, divided by can be rewritten using the rule as follows. This rule can be explained by writing the powers as a product.
This rule is valid for all values of and when Note that the rule might lead to non-real solutions when and at least one of the exponents is not an integer.

## Power of a Power Property

### Rule

When a power is raised to another exponent, the powers can be combined into one by multiplying them. For example can be rewritten with this rule as follows. This rule can be explained by writing the powers as products.

This rule is valid for all values of and when For the rule to be true for it is necessary that both exponents are greater than Note that the rule might lead to non-real solutions when and at least one of the exponents is not an integer.

## Power of a Product Property

### Rule

When the base in an exponential expression is a product raised to some power, the term can be rewritten as a product of two powers. For example, can be rewritten using this rule as follows. This rule can be explained by writing the expression with repeated multiplication.

This rule is valid for all values of when and For the rule to be valid when either or equals it is necessary that Note that the rule might lead to non-real solutions when at least one of the bases is less than and the exponent is not an integer.

## Power of a Quotient Property

### Rule

When the base of a power is a fraction, the term can be rewritten as a fraction of two powers. The rule states that the exponent is applied to the numerator and the denominator. For example, can be rewritten using this rule as follows. This rule can be explained by writing the power with repeated multiplication.
For this rule to be valid it is necessary that Also, for the rule to be valid when it is necessary that Note that the rule might lead to non-real solutions when at least one of the bases is negative and is not an integer.

## Negative Exponent

### Rule

When an exponent of a power is negative, the term can be rewritten as a fraction, with the base in the denominator. Note that the sign of the exponent is changed. For example, can be rewritten with this rule as follows. This rule can be explained by writing the exponent as and applying the Quotient of Powers Property.
This rule is valid for all values of and when Note that the rule might lead to non-real solutions when and is not an integer.

## Zero Exponent

### Rule

Powers with an exponent of are equal to This can be explained by writing an exponent of as a difference, for instance and using the Quotient of Powers Property. Consider

Rewrite as

This rule is valid for all values

## Variable Exponents

The properties of exponents are not only valid for numerical exponents. They can be applied in the same way when exponents contain variables.
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Exercise

Simplify the following expression using the properties of exponents.

Show Solution
Solution
To begin, we notice that each power in this expression has the same base. We will first focus on simplifying the numerator by applying the Product of Powers Property.
We can now simplify the fraction using the Quotient of Powers Property.
The given expression simplifies completely to
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