# Exponent Notation: Definition and Rules

In this lesson, you will learn about the exponent notation, its components and how to calculate it, as well as extract the most important properties of powers used to simplify algebraic expressions. Power, quite simply, is multiplying a number of times by itself. For example, we take the power as the product result of , where the small, raised number to the right of the base number indicates the number of times the base number is multiplied by itself.

### Note:

Let a real number and a whole positif number, then the nth power of is:

The number is called the base, and is called the exponent.

#### Example1:

Let's calculate the following powers:

Now lets do the inverse, transform the follwing numbers to exponnent notation:

## Exponential Notation Rules

### The Product Rule

From the above definition and examples, we can easily calculate the product of the two powers. we've got:

How do you think we got the power that the exponent is ?

You will no doubt have noticed that if we add the exponents of the powers and , we get 6. From this example, we can deduce the following property:

### Note:

Let a real number and and are integers. We have:

When multiplying exponential expressions with the same base, add the exponents. Use this sum as the exponent of the common base.

#### Example2:

Multiply each expression using the product rule:

### The Quotient Rule

From the above definition and examples, we can easily calculate the quotient of the two powers. we've got:

How do you think we got the power that the exponent is ?

In this example, we have subtracted the exponent of the power in the denominator from the exponent of the power in the numerator (). From this example, we can deduce the following property:

### Note:

Let a non-zero real number and and are integers. We have:

When dividing exponential expressions with the same nonzero base, subtract the exponent in the denominator from the exponent in the numerator. Use this difference as the exponent of the common base.

#### Example3:

Divide each expression using the quotient rule:

### The Products-to-Powers Rule for Exponents

In this section, the rule of exponents applies when we are raising a product to a power. Let's take the following example:

The base of the power is the product .

applying the formula in the definition:

Now, how did we get the answer ?

If you notice, we've removed the parentheses and raised both factors, and , to the fourth power. From this example, we can deduce the following property:

### Note:

Let and are real numbers and an integer. We have:

When a product is raised to a power, raise each factor to that power.

• Simplify:
• Simplify:

### The Quotients-to-Powers Rule for Exponents

In this section, the rule of exponents applies when we are raising a quotient to a power. Let's take the following example:

The base of the power is the quotient .

applying the formula in the definition:

Now, how did we get the answer ?

If you notice, we've removed the parentheses and raised denominator and numerator, and , to the fourth power. From this example, we can deduce the following property:

### Note:

Let and are real numbers with and an integer. We have:

When a quotient is raised to a power, raise the numerator to that power and divide by the denominator to that power.

• Simplify:
• Simplify:

### The Negative-Exponent Rule

In the definition of the exponontial notation, We saw that the exponent is a positive integer, but what about if you see a negative exponent? We can easily calculate a power with negative exponent follwing the below example: We simplify the quotient \frac{5^3}{5^5} with two different ways.

This means that . A negative exponent in the numerator becomes a positive exponent when moved to the denominator and vice versa.

### Note:

Let be a nonzero real number and be any integer.

#### Example6:

Write each expression without negative exponents, and evaluate if possible

### The Zero-Exponent Rule

There is a special case in calculating powers, when the exponent is zero. How can we calculate a power taht zero as an exponent? As in the previous paragraphs, we start from the following two examples:

This means that

### Note:

Let a nonzero real number.

#### Example7:

Here are examples involving simplification using the zero-exponent rule:

### Things to remember about exponents

There are some special cases which are very important in simplifying algebraic expressions. We can prove these special cases in the same way in the previous proofs.

### Note:

1. Any base without an exponent has an “invisible” exponent of 1.

2. Any power with a base 1, equals 1.

3. if the exponent is an even number:

4. if the exponent is an odd number: