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# Number Power

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Last updated date: 16th Jul 2024
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## 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 xn

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 ${5^3}$

${5^3} = 5 \times 5 \times 5 = 125$