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# Exponent|Definition & Meaning

## Definition

The definition of **Exponent** is the method of representing large **numbers** in terms of **powers.** This means that the **exponent** is how many **times** a number multiplied by **itself.**

We **sometimes** read numbers in words such as **hundred, thousands,** up to **crores,** and so on. What happens if a number has more digits than we can **read?** For example, 5972190000000000000000000 kg is the **mass** of **Earth.** This number cannot be read in simple **words.** Thus, making it easy to **pronounce** these types of **numbers,** we put the exponents to **work. **

**Exponentiation** is a mathematical process, written as b^{n}, which **includes** two numbers, the **base b** and the **exponent** or **power n**, and pronounced as “**b (raised) to (the power of) n**“. When n is a positive integer. In the above **figure,** b is 6, and n is 2. **Exponentiation** corresponds to repeated **multiplication** of the base b, and b^{n} is the **product** of **multiplying** base n times:

For **example,** 5 is **multiplied** by itself 3 times, that makes, 5 x 5 x 5. This can also be **written** as 5^3. Here, 5 is the **base,** and 3 is the **exponent.** This can also be **read** as 5 is raised to **power** 3.

## Exponent Symbol

The **symbolic** representation for the **exponent** is ^. The name of this symbol (^) is a carrot. For **example,** 8 raised to the times 2 can be **written** as 8^{2}. Thus, 8^{2} = 8 x 8 = 64. The **table** below shows the model of a few numerical **expressions** using **exponents.**

## Exponent Laws

Other **laws** of exponents are mentioned below, which are based on the **powers** they bear.

**Multiplication Law:** Apply on bases **multiplying** the same ones, add the exponents or powers and the base should be **kept** the same. When **bases** are raised with power to another, keep the base the **same** and **multiply** the exponents.

**Division Law:** Apply on **bases** dividing the same ones; **subtract** the power of the **denominator** from the power of the **numerator** while keeping the base the **same.**

Let’s say **‘p’** is any **integer** or a decimal number and **‘a’**, and** ‘b’** are positive integers, that means the **powers** to the bases such that the above laws can be **written** as:

- p
^{a}. p^{b}= p^{a+b} - (p
^{a})^{b}= p^{ab} - (pq)
^{b}= p^{b}q^{b} - (p/q)
^{b}= p^{b}/q^{b} - p
^{a}/p^{b}= p^{a-b} - p
^{a}/p^{b}= 1/p^{b-a}

These **laws** show us the above **properties** of exponents. These laws are **used** to write large numbers in an **understandable** manner and to **simplify** complex algebraic **expressions.**

## Exponent and Powers

As **described** above, the **exponent** is the number of times a number is **multiplied** by itself. The **power** is an expression that tells us about the repeated **multiplication** of the same number. For **example,** in the expression 5^{3}, 3 is the **exponent,** and 5^{3} is **called** the 5 to the power of 3. This means, 5 is **multiplied** by itself 3 times.

## Negative Exponents

A **negative** exponent is **handled** differently in exponents. In the Negative exponent, the **multiplication** of reciprocals of bases takes place. The **multiplication** of reciprocals is done n times (exponent value). For instance, if it is **provided** that a^{-n}, it can be written in its **reciprocal** form as 1/a^{n}. It indicates we have to multiply 1/a ‘n’ times.

Negative **exponents** are utilized when noting **fractions** with exponents. Few samples of negative exponents are 2 x 3^{-9}, 7^{-3}, 67^{-5}, etc. We can transform these **fractions** into positive exponents as:

- 2 × 3
^{-9}= 2 × (1/3)^{9}= 2 / 3^{9} - 7
^{-3}= 1 / 7^{3} - 67
^{-5}= 1 / 67^{5}

## Exponents With Fractions

In a **fractional** exponent, the exponent of a base digit is a fraction. Mathematical operations like cube roots, square roots, or any n^{th} root are instances of fractional **exponents.**

The base with power 1/2 is **named** the square root of that base. Likewise, a base with an **exponent** of 1/3 is named the cube root of that base. Some other illustrations of fractional **exponent** are 5^{2/3}, -8^{1/3}, 10^{5/6}, etc. We can **correspond** these as:

- 5
^{2/3}= (5^{2})^{1/3}= 25^{1/3}= $\mathsf{\sqrt[3]{25}}$ - -8
^{1/3}= ((-2)^{3})^{1/3}= -2 - 10
^{5/6}= (10^{5})^{6}= $\mathsf{\sqrt[6]{10^5}}$ = $\mathsf{\sqrt[6]{100000}}$

## Decimal Exponent

A **decimal** exponent is an exponent in which the **power** of a number is decimal. It is a little **challenging** to calculate the right answer to decimal **exponents,** so we find the approximate solution in such scenarios. Decimal exponents are **evaluated** by first **shaping** the exponent from decimal to **fraction** form. For instance, 4^{1.5} can **correspond** to 4^{3/2}, which can be again simplified to obtain the last **answer,** 8. That is, 4^{3/2} = (2^{2})^{3/2} = 2^{3} = 8.

## Scientific Notation With Exponents

For **noting** very big numbers or very small **numbers,** scientific notation is used as the Standard form. In the standard form, the numbers are noted with the **assistance** of 10’s powers and base decimals. A **number** is denoted in scientific notation if it is between 0 to 10 and multiplied by an **exponent** of 10. The exponent of 10 will be positive if the **numeral is greater than 1**, **while** the power of 10 will be negative if the number is **less than 1. **The below steps will explain how to write **numbers** in scientific notation using **exponents.**

**Step 1:**After the first**integer**of the numeral from the left, place a decimal point. we don’t**require**to place a decimal in a**number**if there is only one integer in a numeral except for zeros.**Step 2:**Take the**product**of that number with an exponent of 10 so that the exponent number**becomes**equal to the times we move the**decimal**point in the base.

By **observing** these two easy steps, we can transform any **numeral** in standard utilizing **exponents,** for instance, 560000 = 5.6 x 10^{5}, and 0.00736567 = 7.36567 x 10^{-3}.

**A Solved Example Involving Exponents**

**Simplify** the below expression:

(3^{2} x 3^{-5}) / 9^{-2}

**Solution**

In the **first** step, shift 9^{-2} to the numerator by taking the **negative** of the exponent.

= (3^{2} x 3^{-5}) x 9^{2}

In **second** step add 3^{2-5} and re-write 9 as 3^{2}:

= 3^{-3} x (3^{2})^{2}

**Multiply** the exponents 2 x 2 = 4:

= 3^{-2} x 3^{4}

In the **third** step, add the exponents as the **bases** are the same.

= 3^{-2+3}

= 3^{2}

= 9

*All images/mathematical drawings were created with GeoGebra.*