# How to Multiply and Divide Exponents?

Exponents and powers are used to simplify the representation of very large or very small numbers. Power is a number or expression that represents the repeated multiplication of the same number or factor. The value of the exponent is the number of times the base is multiplied by itself.

Example for exponents, If we need to express 3 × 3 × 3 × 3 × 3 in a simple way, we may write it as 3^{5}, where 3 is the base and 5 is the exponent. The entire expression 3^{5} is considered to represent power. Example for powers:** **5^{3} = 5 raised to power 3 = 5 × 5 × 5 = 125, 6^{4} = 6 raised to power 4 = 6 × 6 × 6 × 6 = 1296. A number’s exponent represents the number of times the number has been multiplied by itself. 3 is multiplied by itself for n times, 3 × 3 × 3 × 3 × …n times = 3^{n}. 3^{n} is an abbreviation for 3 raised to the power of n. As a result, exponents are sometimes known as power or, in certain cases, indices.

### General Form of Exponents

The exponent indicates how many times a number should be multiplied by itself to obtain the desired results. As a result, any number ‘b’ raised to the power ‘p’ may be expressed as :

b^{p}= {b × b × b × b × … × b} p times

Here b is any number, and p is a natural number.

- Here, b
^{p}is also called the p^{th}power of b. - ‘b’ represents the base, and ‘p’ is the exponent or power.
- Here ‘b’ is multiplied ‘p’ times, and thereby exponentiation is the simplified method of repeated multiplication.

**Some basic rules of Exponents **

- Product Rule ⇢ a
^{n}× a^{m}= a^{n + m}- Quotient Rule ⇢ a
^{n}/ a^{m}= a^{n – m}- Power Rule ⇢ (a
^{n})^{m}= a^{n × m}or^{m}√a^{n}= a^{n/m}- Negative Exponent Rule ⇢ a
^{-m}= 1/a^{m}- Zero Rule ⇢ a
^{0}= 1- One Rule ⇢ a
^{1}= a

### How to Multiply and Divide Exponents?

**Solution:**

Exponents and powers are used to simplify the representation of very large or very small numbers.

**To Divide Exponents**

The laws of exponents simplify the process of simplifying expressions. When dividing exponents with the same base, the basic rule is to subtract the given powers. This is also known as the Division Law or Exponent Quotient Property.

**m ^{n}_{1} ÷ m^{n}_{2} = m^{n}_{1}/ m^{n}_{2} = m^{(n}_{1}^{ – n}_{2}^{)}**

**First Case: Dividing Exponents with the Same Base **

We utilize the basic rule of subtracting the powers to divide exponents with the same base. Consider the expression m^{n}_{1} ÷ m^{n}_{2}, where ‘m’ is the common base and the exponents ‘n_{1}‘ and ‘n_{2}‘ are the exponents. According to the ‘Quotient property of Exponents,’

**m ^{n}_{1} ÷ m^{n}_{2} = m^{n}_{1}/ m^{n}_{2 }= m^{(n}_{1}^{ – n}_{2}^{) } **

**Example: Divide 3 ^{5} ÷ 3^{3} **

Here as we can see bases are same but different powers .

So the division law or Quotient law : m

^{n}_{1}÷ m^{n}_{2}= m^{n}_{1}/ m^{n}_{2}= m^{(n}_{1}^{ – n}_{2}^{) }Here, 3

^{5}÷ 3^{3}= 3

^{5}/3^{3}= 3

^{(5-3)}= 3

^{2}

**Second Case: Dividing Exponents with different Bases **

We apply the ‘Power of quotient property’ to divide exponents with different bases and the same exponent, which is

(m/n)^{p}= m^{p}/n^{p }

Consider the formulas m^{p} ÷ n^{p}, which has distinct bases but the same exponent.

**Example: Divide: 15 ^{3} ÷ 3^{3}.**

This can be solved using the ‘Power of quotient property’ as,

(m/n)

^{p}= m^{p}/n^{p}.= 15

^{3}÷ 3^{3}= (15 / 3)

^{3}= 5

^{3}.

**To Multiply Exponents **

**First Case: When Multiply exponents with the same Base **

According to this rule: The product of two exponents with the same base but distinct powers equals the base raised to the sum of the two powers or integers; this is also known as the Multiplication Law of Exponents. When multiplying two expressions with the same base, we can use,

m^{n}_{1}× m^{n}_{2}= m^{(n}_{1}^{ + n}_{2}^{)}

Where m is the common base and n_{1 }and n_{2}are the exponents.

**For Example, Multiply 3 ^{3} × 3^{6}?**

Given: 3

^{3 }× 3^{6}Here bases are same. So we will use: m

^{n}_{1}× m^{n}_{2}= m^{(n}_{1}^{ + n}_{2}^{)}Therefore, = 3

^{(3+6)}= 3

^{9}

**Second Case: When Multiply Exponents with a different bases**

When there is different base with same exponents , we will use the formula :

m^{p}× n^{p}= (m × n)^{p}.

Here m and n are the different bases and p is the exponent.

**Example: Multiply 2 ^{3} × 4^{3}**

Given: 2

^{3}× 4^{3}Here, we will use: m

^{p}× n^{p}= (m × n)^{p}= (2 × 4)

^{3}= 8

^{3}

In these ways in different cases we can divide and multiply Exponents.

### Sample Questions

**Question 1: Simplify or Divide 25 ^{4}/5^{4} **

**Solution: **

Here bases are different with same Exponent,

We will use the formula, (m/n)

^{p}= m^{p}/n^{p }Therefore, = 25

^{4}/5^{4}= (25/5)

^{4}= 5

^{4}= 625

**Question 2: Find the value of the expression, 15 ^{8} × 15^{3}**

**Solution:**

Given: 15

^{8 }× 15^{3}When multiplying two expressions with the same base but different exponent,

m

^{n}_{1}x m^{n}_{2}= m^{(n}_{1}^{ + n}_{2}^{)}formula, where m is the common base and n_{1}and n_{2}are the exponents.By Applying this rule,

we get, = 15

^{8 }× 15^{3}= 15

^{(8 + 3)}= 15

^{11}

**Question 3: What is the product of (2x ^{3}y^{5} ) and (3x^{4}y^{2})?**

**Solution:**

The product of (2x

^{3}y^{5}) and (3x^{4}y^{2})= (2x

^{3}y^{5}) × (3x^{4}y^{2})= (2 × 3) × x

^{3}x^{4}× y^{5}y^{2}When multiplying two expressions with the same base, we can use

mformula, where m is the common base and n^{n}_{1}× m^{n}_{2}= m^{(n}_{1}^{ + n}_{2}^{)}_{1 }and n_{2}are the exponents.= 6x

^{3+4}× y^{5+2}= 6x

^{7}y^{7}

**Question 4: What is x ^{3} divided by x^{2}?**

**Solution:**

Here given: x

^{3}divided by x^{2}here bases are same but exponents are different,

So we use the division law or Quotient law: m

^{n}_{1}÷ m^{n}_{2 }= m^{n}_{1}/ m^{n}_{2}= m^{(n}_{1}^{ – n}_{2}^{) }So write it as x

^{3}/x^{2}= x

^{3 – 2}= x

^{1}= x

**Question 5: Evaluate a ^{3} × a^{5 }× a^{-6} **

**Solution:**

Given that: a

^{3}× a^{5}× a^{-6}Here bases are same but exponents are different ,By using product rule or multiplication law .

m

^{n}_{1 }× m^{n}_{2}= m^{(n}_{1}^{ + n}_{2}^{)}= a

^{3}× a^{5}× a^{-6}= a

^{(3 +5)}× a^{-6}= a

^{8}× a^{-6}= a{8+ (-6)} {Using by product rule}

= a

^{8-6}= a

^{2}

**Question 6: Divide 10 ^{5}/5^{5}**

**Solution:**

Here bases are different with same Exponent ,

we will use the formula : (m/n)

^{p}= m^{p}/n^{p}Therefore, = 10

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

^{5}= 5

^{5}= 3125