## Exponents and Powers

Exponents and Powers are mathematical operations used to represent and represent large sums of numbers or minimal numbers in a simplified manner. Students know the basics of how to calculate an expression 5 x 5, but, this expression can be simplified shortly and crisply using a concept called the exponents. An expression that represents the repeated multiplication of the same factor is known as the power.

For example, the sum 3 x 3 x 3 x 3 is an equation, but a simple way to write it is 34, where 4 is the exponent power and 3 is the base number. The full operational expression 34 is said to be the power.

Power is an expression that presents the repeated multiplication of the same factor or number. The exponent value is based on the repeated number of times that the base is multiplied to itself.

### Exponent Formula

From the expression a2, 'a' is known as the base number, and the two is referred to as the exponent. The exponent number represents the number of times the base number is to be multiplied.

a\[^{m}\] . a\[^{n}\] = a\[^{m+n}\]

(a\[^{m}\])\[^{n}\] = a\[^{mn}\]

For example, in a2, the base number ‘a’ will be multiplied twice, that is, a x a and similarly, a3 = a × a × a. Here are a few of the important exponent form-

a0 =1

a1= a

√a=a1/2

n√a=a1/n

a−n = 1/an

an = 1/a−n

aman=am+n

am/an=am−n

(am)p=amp

(amcn)p=ampcnp

(am/cn)p=amp/cnp

## Laws of Exponents

There are three basic exponent laws that every student needs to comprehend. Each rule of exponents help students solve different types of mathematical equations and teach them basic concepts like addition, subtraction, multiplying, and even division exponents.

Law #1: a\[^{n}\] x a\[^{m}\] = a\[^{n+m}\]

Law #2: \[\frac{a^{m}}{a^{n}}\] = a\[^{m-n}\]

Law #3: (a\[^{n}\])\[^{m}\] = a\[^{n \times m}\]

The laws of exponents are demonstrated based on the powers each expression carries.

### 1. Division Law

The division law is applicable when two exponents have the same base numbers but different powers. The expression is divided, thus, resulting in the base number raised to the difference between the two power numbers.

An example of dividing exponents: am ÷ an = am / an = am - n

### 2. Multiplication Law

The Multiplication law is applicable when the product of two exponents comprise the same base numbers but comprise different powers. The result of the multiplying exponents equals the base raised to the sum of the two integers or powers.

An example of multiplying exponents: am × an = am + n

### 3. Negative Exponent Law

The Negative Exponent Law is applicable when any base numbers comprise a negative power. This expression results in the reciprocal but with the positive integer or positive to the base number.

An example of negative exponent law: a - m = 1/am

## Exponent Rules

The exponent laws follow the exponent rules. There are four basic exponent rules to follow-

Let's consider the following- suppose the integer values are 'a' and 'b' and the power values are 'm' and 'n', then the rules of exponent and power are as follow-

### 1. (ax)y = a(xy)

The expression results as- 'a' raised to the power of 'x' presented to the power 'y' outputs to 'a' submitted to the power of the product numbers of 'x' and 'y'.

The example of this expression is (52)3 = 52 x 3

### 2. ax/bx = (a/b)x

The expression results as- the division of the 'a' raised to the power of 'x' and 'b' presented to the power of 'x' outputs to the division of 'a' and 'b', the whole raised to the power 'x'.

The example of this expression is 52/62 = (5/6)2

### 3. a0 = 1

As per this exponent rule, if the power of any integer is zero, the output of the expression leads to one or unity.

The example of this expression is 50 = 1

### 4. ax × bx =(ab)x

The expression results as- the product of the expression 'a' raised to the power 'x' and 'b' presented to the power 'x' outputs to the product of 'a' and 'b' the whole raised to the power 'x'.

Example is 52 × 62 =(5 x 6)2

Q1. State the Properties of Exponents.

Ans. The five properties of exponents-

Power to a Power - This property is based on the fact that the Quotient undergoes division of the two powers with the same base and then subtracting the exponents.

Power of a Product - This property is based on the fact that the product of a power, results in each base of its exponent.

Product of Powers - This property is based on the fact that numbers are multiplied exponentials with the same base and then add their exponents.

Quotient of Powers - This property is based on the fact that the division of exponents with the same base and then subtract the exponent.

Power of a Quotient - This property is based on the fact that the power of the Quotient results in the Quotient where the numerator and the denominator are raised to the power separately, and then the whole expression is divided.

Q2. Define Exponents and Enlist the Seven Laws of an Exponent.

Ans. Exponents are mathematical operations that represent large sums of numbers or minimal numbers in a simplified manner.

The seven laws of exponent are-

Raise Quotient to a Power: Distribute the power over each term in the Quotient.

Divide Powers of the Same Base: This law applies to the bases that are the same, then subtract the exponent.

Zero Exponents: This law applies to the integer to the zero power besides 0, results to 1. Raise power to a power: Multiply the exponents together and then add the exponents. Raise product to power: The product is raised to a power, distributing the power to each term in the product.

Raise a Power to a Power: Multiply the powers with the same base.

Negative Exponents: If an exponent is a negative put the integer under 1 for division, and it turns positive.