## Introduction

A base number being raised to an exponent is referred to as a "power" in mathematics. It signifies that "base number" and "exponent" are the two fundamental components of powers.

## Number Power:

How to find the power of a number? How many times to multiply a given number depends on its power (or exponent).

It appears as a small number above and to the right of the basic number.

The small "2" in this illustration tells us to multiply 8 twice:

\[{8^2} = 8 \times 8 = 64\]

Hence the Number Power Formula for 8 times 8 will be \[{8^2}\].

However, the term "power" can also refer to the outcome of an exponent, so in the previous example, "64" is also referred to as the power.

Different Terms of Exponent

## Exponent of a Number:

A number's exponent indicates how many times the number has been multiplied by itself. Example: Since 2 is multiplied by itself 4 times, \[2 \times 2 \times 2 \times 2\] can be expressed as \[{2^n}\]. Here, 4 is referred to as the "exponent" or "power," while 2 is referred to as the "base." Generally speaking, \[{x^n}\] denotes that x has been multiplied by itself n times.

Exponent of a Number

Here, Here, in the term x^{n}

x is called the "base"

n is called the "exponent"

And is read as "x to the power of n" or "x raised to n".

## Power of Exponents:

We may easily express and represent extremely big and small numbers using exponential notation or the exponential form of numbers. For instance, 10000000000000 is equal to \[1 \times {10^{13}}\] whereas 0.0000000000000007 is equal to\[7 \times {10^{ - 16}}\]. This makes numbers easier to read, helps in ensuring their accuracy, and saves us time.

## Properties of Exponents:

These characteristics are regarded as major exponents rules that must be followed when solving exponents. The following list includes exponent qualities.

Product law: \[{a^m} \times {a^n} = {a^{m + n}}\]

Quotient law: \[{a^m} \times {a^n} = {a^{m -n}}\]

Zero Exponent law: \[{a^0} = 1\]

Negative Exponent law: \[{a^{ - m}} = 1 \div {a^m}\]

Power of a Power law: \[{({a^m})^n} = {a^{mn}}\]

Power of a Product law: \[{\left( {ab} \right)^m} = {a^m}{b^m}\]

## Important Points for Power:

When a fraction's exponent is negative, we take the fraction's reciprocal to make it positive. Consequently, \[{\left( {\frac{a}{b}} \right)^{ - m}} = {\left( {\frac{b}{a}} \right)^m}\]

We can set the bases to equal when the exponents in an equation are the same on both sides, and vice versa.

## Solved Examples:

1: Each tree in a garden has roughly \[{5^7}\] leaves, and there are about \[{5^3}\] trees overall. Calculate the total number of leaves using exponents.

Ans: The number of trees in the garden is \[{5^3}\] , and each tree has \[{5^7}\] leaves. \[{5^3}\]\[ \times \] \[{5^7}\] = \[{5^{10}}\] leaves total, according to the exponents law.

Therefore, there are \[{5^{10}}\] leaves in all.

2: What is 2 when it has a 7 exponent?

Ans: When 2 has an exponent of 7 then the answer will be \[{2^7} = 128\].

As \[2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2\] \[ = 128\].

3: Solve \[{25^3}/{5^3}\].

Ans: Using quotient Law: \[{a^m} \div {a^n} = {a^{\frac{m}{n}}}\]

It is possible to write \[{25^3}/{5^3}\] as \[{(25/5)^3}\]

\[ = {5^3}\]

## Conclusion:

An expression known as "power" denotes the process of repeatedly multiplying a value or integer. \[{a^n}\] is often a power where n is the exponent and \[{a^n}\] is the base.

## FAQs on Number Power

1. How do negative exponents work?

When 1 is divided by a component that has been multiplied repeatedly, a negative exponent is employed. Let's say that \[{n^{ - 1}}\], where -1 is the exponent, gives 1/n. A number shows its reciprocal when it is increased to negative exponents. For instance, \[{3^{ - 2}}\] , or \[1/{3^2}\]

2. If the exponent is 1 or 0, what do we get?

The value of the base remains unaltered if the exponent of a base number is one. For instance, \[{9^1} = 9\].

In the event that the exponent is 0, the result is 1. For instance, \[{9^0} = 1\].

3. Give one example of exponent.

One example of exponent is \[3 \times 3 \times 3 \times 3 = {3^4} = 81\].

4. What is the number power formula for 5 raised to 3?

5 raised to 3 means \[{5^3}\]

\[{5^3} = 5 \times 5 \times 5 = 125\]