Exponent Rules are used to represent very large numbers or very small numbers. The laws of exponents are used to solve different problems of exponents. Multiplication, division, etc. can be done on exponents according to these laws. There are various rules of exponents, also known as laws of exponents, in Mathematics. All these laws are mentioned in the following article: Exponents Definition, Laws of Exponents, Examples of Exponents and others in detail.

## Exponents Definition

When a number is raised to some power then the power on the base number is called Exponent. Exponent simply means a base number is multiplied by itself equal to the power mentioned on it.

For example, if we say P^{n }it means **P** is divisible by ‘**n**‘ a number of times. It can be written as P×P×P×P×P×P . . . n times.

When the exponent is 2, it is also called ‘squared,’ while when the exponent is 3, it is called ‘cube.’ When we are multiplying the area, we multiply the length (m / cm) by 2, and when we are multiplying the volume (unit = m / cm), we are multiplying it by 3.

Exponent allows us to write large numbers as well as small numbers. For example, you can write the mass of the Earth, which is 5.97219 x 1024 kg, or the mass of the electron, which is 9,1 x 10-31 kg.

## What are Exponent Rules?

Laws of Exponent Rules are the set of rules that help us to solve mathematics problems in an easy way. Since we can get getter size exponents that make multiplication long then with the help of laws of exponents, we can solve the problems easily and in a time-bound manner.

Familiarity with the seven key Laws of Exponents that we must know to solve arithmetic problems involving exponents:

**b**^{m}** × **b^{n}** = b**^{(m+n)}

** Also, Read:** Geometry Formulas for 2D and 3D

## What are the Laws of Exponents?

Laws of Exponents are the set of rules that help us to solve mathematics problems in an easy way. Since we can get longer exponents that make multiplication longer then with the help of laws of exponents, we can solve the problems easily and in a time-bound way.

Familiarity is the seven Laws of Exponents that we must know to solve arithmetic problems involving exponents:

- Product of Powers Rule
- Quotient of Powers Rule
- Power of a Powers Rule
- Power of a Powers Rule
- Power of a Quotient Rule
- Zero Power Rule
- Negative Exponent Rule

**Product of Powers Rule**

In the **Product of Powers** **Rule**, if two numbers with the same bases and different exponents are multiplied then exponents of the base are added to find the product. It is represented as x^{m}×x^{n} = x^{(m+n)}

**Example:** **7 ^{2} × **7

^{3}=?7

^{2}

*×*7

^{3}

*=*7

^{2+3}

*=*7

^{5}

multiply five by itself five times.

7

^{5}= 7 × 7 × 7 × 7 × 7 = 16807

**Quotient of Powers Rule**

In **Quotient of Powers** **Rule**, if two numbers with the same bases and different exponents are divided then the exponents of the base are subtracted to find the quotient. It is represented as x^{a}÷x^{b} = x^{(a-b)}

**Example: 4 ^{5} ÷ 4^{3} =?**

4

^{5}÷ 4

^{3}= 4

^{5-3}= 4

^{2}

if necessary, simplify the equation.

4

^{2}= 4 × 4 = 16

**Power of a Power Rule **

In **Power of a Power** **Rule**, if a number raised to some power is again raised to some power then the two powers will be multiplied. It is represented as (x^{m})^{n} = x^{m×n}

**Example:** **(2 ^{3})^{2}=?**

Multiply the exponents together in equations like the one above while keeping the base constant.

2

^{3×2}= 2

^{6}

**Power of a Product Rule**

In the **Power of a Product** **Rule**, two different bases are raised to the same power are multiplied, then, bases are multiplied and power is common to the product of the bases. It is represented as (x^{m} × y^{m}) = (xy)^{m}. If the given question is (xy)^{m} then distribute the exponent to each portion of the base when multiplying any base by an exponent, hence (xy)^{m} = (x^{m} × y^{m})

**Example: 2 ^{3}×3^{3} =?**

Since the bases are different and the power is the same multiply the bases and raise it to the common power.

Therefore, 2

^{3}×3

^{3}=(2×3)

^{3}= 6

^{3}= 216

**Power of a Quotient Rule**

In **Power of a Quotient Rule**, if two different bases with the same power are divided then the result is the quotient of the bases raised to the same power. This is represented as x^{m}/y^{m}= (x/y)^{m}. In this case, vice versa is also true i.e. if both numerator and denominator are raised to the same power then power is distributed to both numerator and denominator individually. It can be represented as (x/y)^{m} = x^{m}/y^{m}

**Example:** **Simplify 6 ^{4}/3^{4}**

In this case, find the quotient of the bases and raise common power to it.

6

^{4}/3

^{4}= (6/3)

^{4}= 2

^{4}= 16

**Zero Power Rule**

In the **Zero Power Rule**, if any base is raised to power zero, then the result will be 1. This can be represented as x0 = 1. Zero Power rule can be understood from the following description

Suppose we have to prove x^{0} = 1.

x^{0} = x^{n-n} , where (0 = n-n)

From the Quotient of Power Rule, we know that if the base are same then we subtract the exponents while finding the quotient; the vice versa of Quotient of Power Rule also holds true.

⇒ x^{n-n} = x^{n}/x^{n} = 1

Hence, x^{0} = 1.

**Negative Exponent Rule**s

In the **Negative Exponent Rule**s, if a number is raised to negative interest then we convert the base to its reciprocal, and the power is changed to positive. The vice versa is also true i.e. if the exponent is positive and if the base is converted to its reciprocal then the exponent is changed to the negative value. It can be represented as (x/y)^{-m} = (y/x)^{m}

**Example: (2/3) ^{-2} =?**

the exponent is negative the base is converted to its reciprocal.

⇒ (2/3)

^{-2}= (3/2)

^{2}= 3

^{2}/2

^{2}= 9/4

## Laws of Exponents and Logarithms

The Laws of Exponents and the Logarithm Rules are 2 rules that are used to solve various mathematical problems and these rules are added in the table below.

Rules | Exponnets | Logarithms |
---|---|---|

Product Rule | x^{p}.x^{q} = x^{(p+q)} | log_{a}(mn) = log_{a}m + log_{a}n |

Quotient Rule | x^{p}/x^{q} = x^{(p-q)} | log_{a}(m/n) = log_{a}m – log_{a}n |

Power Rule | (x^{p})^{q} = x^{p.q} | log_{a}mn = n log_{a}m |

## Exponent Rules Examples

**Example 1: What is the simplification of **5^{3} ×5^{1}?

**Solution:**

5^{3} × 5^{1} = 5^{3+1} = 5^{4}

**Example 2: Simplify and find the value of 10**^{2}**/5**^{2}**.**

**Solution:**

We can write the given expression as;

10^{2}/5^{2}= (10/5)^{2} = 2^{2} = 4

**Example 3: Find the value of (256)**^{3/4}

**Solution:**

(256)^{3/4} = (4^{4})^{3/4} = 4^{4×(3/4)} = 4^{3} = 64

**Example 4: Find the value of 7**^{-3}

**Solution:**

7^{-3} = (1/7)^{3} = 1^{3}/7^{3} = 1/343