# Exponents and Powers Formulas

Can we read 10,000,000,000,000,000? These massive natural numbers are not easy to read as well as to recognize and hence evaluate. Exponents make it easy to read and handle very large numbers. We also call exponents, powers or indices in mathematics. Exponents and powers make the complex computations easy and faster. In this topic, we will discuss various exponents and powers formulas with examples. Also, the laws of exponents will be discussed. Let us learn it!

## Exponents and Powers Formulas

### What are Exponents and Powers?

We know how to calculate the expression like 6 × 6. This expression can be written in a shorter way using the exponents.So, $$6 \times 6 = 6 ^ 2$$. Also, $$5 \times 4 = 5 ^ 4$$

An expression that represents repeated multiplication of the same factor is power. The number 6 is the base, and the number 2 is its exponent. The exponent corresponds to the number of times the base will be multiplied by itself.

Therefore, if two powers have the same base then we can multiply these two powers. When we multiply two powers, we will add their exponents. If two powers have the same base then we can divide the powers also. When we divide these powers then we subtract their exponents. Here in this topic, the student will learn about various formulas of exponents.

When we need to repeatedly multiply some number by itself, then it means that we raise it some power. We know this as the Exponent. The power in the exponent represents the number of times that we want to carry out the multiplication operation. Exponents are having their own set of rules when it comes to carrying out Arithmetic Operations. These formulas will help a lot with many mathematical simplifications.

Source:en.wikipedia.org

### The Exponents and Powers Formulas are

Laws of exponents are used to simplify the calculations. Some important laws of exponents are:

1. Power zero $$a^{ 0 } = 1$$
2. Power one $$a^{ 1 } = a$$
3. Fraction formula $$\sqrt{ a } = a^{ \frac{ 1 }{ 2 }}$$
4. Reverse formula : $$\sqrt{ n } { a } = a^{\frac{ 1 } { n }}$$
5. Negative power value : $$a^{ -n } = \frac{ 1 }{ a^{n} }$$
6. Fraction formula : $$a^{n} = \frac{1}{ a^{ -n } }$$
7. Product formula : $$a^{m}a^{n} = a^{ m + n }$$
8. Division Formula : $$\frac{ a^{ m }}{ a^{ n }} = a ^{ m-n }$$
9. Power of Power formula : $$(a^{ m })^{ p } = a^{ mp }$$
10. Power distribution Formula : $$(a^ { m }c^{ n })^{ x } = a ^ { mx } c ^{ nx }$$
11. The Power distribution Formula : $$\left ( \frac {a ^{ m }}{c^{ n }} \right )^{x} = \frac{a^{ mx }}{c^{ nx }}$$

## Solved Examples for Exponents and Powers Formulas

Example-1: Solve $${ 4 }^ { -3 }$$

Solution: As per the the negative exponent rule:

$$\large a^{ -n }=\frac{ 1 }{ a^{n} }$$

i.e. $$\large{ 4 } ^{ -3 }= \frac{ 1 }{ 4 ^ {3} }$$

i.e. $$\large { 4 } ^ {-3} = \frac { 1 }{ 64 }$$

Example-2:  Simplify : $${(x^3 y^4)} ^ 2$$

Solution: We will do its simplification as:

$$(x^{ 3 }\cdot y^{ 4 })^{ 2 } = x^{ 3\cdot 2 }\cdot y^{ 4\cdot 2 }=x^{6}\cdot y^{ 8 }$$

Example-3: Simplify: $$(\frac{4}{5})^5 \times (\frac{5}{6}) ^ 5$$

Solution : We can simplify it by using the laws of exponents.

$$(\frac{ 4 }{ 5 })^5 \times (\frac{ 5 }{ 6 }) ^ 5 \\$$

$$= (\frac{ 4 }{ 5 } \times \frac { 5 }{ 6 }) ^ 5 = (\frac{ 4 }{ 6 }) ^5$$

$$= (\frac{ 2 } { 3 }) ^5$$

Share with friends

## Customize your course in 30 seconds

##### Which class are you in?
5th
6th
7th
8th
9th
10th
11th
12th
Get ready for all-new Live Classes!
Now learn Live with India's best teachers. Join courses with the best schedule and enjoy fun and interactive classes.
Ashhar Firdausi
IIT Roorkee
Biology
Dr. Nazma Shaik
VTU
Chemistry
Gaurav Tiwari
APJAKTU
Physics
Get Started

## Browse

##### Maths Formulas

4 Followers

Most reacted comment
1 Comment authors
Recent comment authors
Subscribe
Notify of
Guest
KUCKOO B

I get a different answer for first example.
I got Q1 as 20.5
median 23 and
Q3 26

Guest
Yashitha

Hi
Same

Guest
virat

yes

## Browse

##### Maths Formulas

Watch lectures, practise questions and take tests on the go.

No thanks.