Exponent helps to solve various algebraic problems. It is used to calculate the extremely large or extremely tiny values. These are also used in computers when describing megabytes, gigabytes and terabytes. Mastering basic exponent rules will make your study of algebra productive and enjoyable. In this post we will discuss exponent definition, rules to evaluate exponents and a lot of examples to perfectly understand the concept.

## What are exponents?

An exponent refers to the number of times a number is multiplied by itself. Or

Exponents shows how many times to multiply a base number by itself. For example, 5^{4 }is telling that **five** is multiply by itself **four** times.

The number in which power is written is known as **base**, while the subscript number above it is known as **exponent** or **index **or** power**.

The power of two is said to be **squared** and the power of three is said to be the **cube**.

In general, we use **a ^{n}** tells to multiply

**a**by itself in

**n**times.

## Notation

We use a base and a power to show the exponent problem.

We also use “^” sign to express exponent.

4^3

There is another method of writing the exponent is known as **Python symbol****.**

In this symbol we use ** to write the exponent.

4**3

All three notations can be written in general.

## Rules of exponent

There are several rules of exponent.

### Multiplication of power

When two bases of same number are multiplies, we use this rule. According to this rule keep the bases same and add the exponents.

**Example: **

Calculate 5^{3 }x 5^{5}

**Solution:**

Since values are in product, we use the multiplication of power rule.

Keep the base value same and add the powers.

5^{3}x5^{5} = 5^{3+5}

= 5^{8}

5^{8}= 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 = 390625

**Example: **

Calculate 54^{31 }x 54^{59}

**Solution:**

Since values are in product, we use the multiplication of power rule.

Keep the base value same and add the powers.

54^{31}x54^{59 }= 54^{31+59}

= 54^{90}

Try out exponents calculator to explore further steps to find exponent.

### Quotient of powers rule

When two bases of same number are divides, we use this rule. According to this rule keep the bases same and subtract the exponents.

**Example: **

Calculate 7^{6}/7^{4}

**Solution:**

Since values are in quotient, we use the quotient of powers rule.

Keep the base value same and subtract the powers.

7^{6}/7^{4 }= 7^{6-4}

= 7^{2}

**Example: **

Calculate 17^{16}/17^{4}

**Solution:**

Since values are in quotient, we use the quotient of powers rule.

Keep the base value same and subtract the powers.

17^{16}/17^{4 }= 17^{16-4}

= 17^{12}

### Power of power rule

When an equation has power is being raised by another power, we use this rule. According to this rule we multiply the powers.

**Example: **

Calculate (19^{6})^{4}

**Solution:**

Since values are in power of power, we use the power of powers rule.

Keep the base value same and multiply the powers.

(19^{6})^{4 }= 19^{6×4}

= 19^{24}

**Example: **

Calculate (109^{16})^{5}

**Solution:**

Since values are in power of power, we use the power of powers rule.

Keep the base value same and multiply the powers.

(109^{16})^{5 }= 109^{16×5}

= 109^{80}

### Power of product rule

When an equation has power of product of base, we use this rule. According to this rule we distribute the exponent to each part of base.

**Example: **

Calculate (9×5)^{4}

**Solution:**

Since values are in power of product, we use the power of product rule.

Keep the base value same and distribute the power.

(9×5)^{4 }= 9^{4}x5^{4}

**Example: **

Calculate (102×105)^{2}

**Solution:**

Since values are in power of product, we use the power of product rule.

Keep the base value same and distribute the power.

(102×105)^{2 }= 102^{2 }x 105^{2}

### Power of a quotient rule

When an equation has power of quotient, we use this rule. According to this rule we distribute the exponent to each part of base.

**Example: **

Calculate (2/5)^{6}

**Solution:**

Since values are in power of quotient, we use the power of quotient rule.

Keep the base value same and distribute the power.

(2/5)^{6 }= 2^{6}/5^{6}

**Example: **

Calculate (211/115)^{6}

**Solution:**

Since values are in power of quotient, we use the power of quotient rule.

Keep the base value same and distribute the power.

(211/115)^{6 }= 211^{6}/115^{6}

### Zero power rule

Any base raised to the power of zero, we use this rule. According to this rule any base having power zero always equal to one.

**Example: **

Calculate (5)^{0}

**Solution:**

Since values are in zero power, we use the zero-power rule.

5^{0 }= 1

**Example: **

Calculate (512)^{0}

**Solution:**

Since values are in zero power, we use the zero-power rule.

512^{0 }= 1

### Negative power rule

When a term has a negative exponent, we use this rule. According to this rule flip the whole term in the division of one or reciprocal to turn the negative value into positive.

**Example: **

Calculate (2)^{-3}

**Solution:**

Since values are in negative power, we use the negative power rule.

2^{-3 }= 1/2^{3}

**Example: **

Calculate (122)^{-23}

**Solution:**

Since values are in negative power, we use the negative power rule.

122^{-23 }= 1/122^{23}

## How to calculate Exponent problems?

In this section, we use examples for the calculation of exponent problems.

**Example 1**

Evaluate **3 ^{3}**.

**Solution**

**Step 1: **write the base multiply by itself in exponent times

**3 ^{3 }**= 3 x 3 x 3

**Step 2: **Multiply

**3 ^{3 }**= 3 x 3 x 3 = 27

**Example 2**

Evaluate **3 ^{3 }x 3^{2}.**

**Solution**

**Step 1: **Apply multiplication rule

**3 ^{3 }x 3^{2}** = 3

^{3+2}

= 3^{5}

**Step 2: **write the base multiply by itself in exponent times

**3 ^{5} = **3 x 3 x 3 x 3 x 3

**Step 3: **Multiply

**3 ^{5} = **3 x 3 x 3 x 3 x 3 = 243

**Example 3**

Evaluate **3 ^{5}/3^{2}.**

**Solution**

**Step 1: **Apply quotient rule

**3 ^{5}/3^{2} **= 3

^{5-2}

= 3^{3}

**Step 2: **write the base multiply by itself in exponent times

**3 ^{3} = **3 x 3 x 3

**Step 3: **Multiply

**3 ^{3} = **3 x 3 x 3 = 27

**Example 4**

Evaluate (**2 ^{3})^{2}.**

**Solution**

**Step 1: **Apply power of power rule

(**2 ^{3})^{2} **= 2

^{3×2}

= 2^{5}

**Step 2: **write the base multiply by itself in exponent times

**2 ^{5} = **2 x 2 x 2 x 2 x 2

**Step 3: **Multiply

**2 ^{5} = **2 x 2 x 2 x 2 x 2 = 32

**Example 5**

Evaluate (**2 x 4) ^{2}.**

**Solution**

**Step 1: **Apply power of product rule

(**2 x 4) ^{2}= **2

^{2 }x 4

^{2}

**Step 2: **write the base multiply by itself in exponent times

**2 ^{2} = **2 x 2

**4 ^{2} = **4 x 4

(**2 x 4) ^{2}** = 2 x 2 x 4 x 4

**Step 3: **Multiply

(**2 x 4) ^{2}** = 2 x 2 x 4 x 4 = 64

**Example 6**

Evaluate (**3/4) ^{2}.**

**Solution**

**Step 1: **Apply power of product rule

(**3/4) ^{2 }= **3

^{2}/4

^{2}

**Step 2: **write the base multiply by itself in exponent times

**3 ^{2} = **3 x 3

**4 ^{2} = **4 x 4

(**3/4) ^{2}** = 3×3/4×4

**Step 3: **Multiply and divide

(**3/4) ^{2}** = 3×3/4×4 = 9/16

= 0.5625

**Example 7**

Evaluate (**8) ^{-3}.**

**Solution**

**Step 1: **Apply negative power rule

(**8) ^{-3} **= 1/8

^{3}

**Step 2: **write the base multiply by itself in exponent times

**1/8 ^{3} = **1/8x8x8

**Step 3: **Multiply and divide

**1/8 ^{3} = **1/8x8x8 = 1/512

= 0.002

Verify the above answer using the online exponent calculator.

## Summary

Now you have come to know that the exponent is not difficult or confusing. Once you have the basic knowledge about rules of exponent you will easily find the exponent.