Hello, friends! Today we will discuss an important concept in mathematics. This concept will form the base of future concepts and enable the students to easily understand them. We will look at the exponential formula or rather the sets of them, study their applications on the basis of the exponent rules.

**Table of content**

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## Exponents

In mathematics, there is a concept of exponents. It is derived from the idea of multiplication. We know that in multiplication we obtain the product of 2 numbers for example 2 × 3 = 6. Now, taking this concept further, let us ask the product of multiplying a number by itself. For example 2 × 2 = 4 or 3 × 3 = 9 and 6 × 6 = 36.

The process of multiplying a number “x” by itself is called as Exponent of the term. The number of times a number ‘x’ is multiplied by itself is called as the power of the term. Suppose if the number ‘x’ is multiplied by itself say ‘y’ number of times, it is denoted by x^{y}** ^{ }**where ‘x’ is called the base and ‘y’ the exponent or the power of the base.

It is read as ‘x’ raised to the power ‘y’, and it implies that the number ‘x’ is multiplied by itself ‘y’ number of times.

**Browse more Topics under Exponents And Powers**

### Square

The process of multiplying a number by itself is called squaring the number also called as ‘square‘ of the number. For a number ‘x’ square of the number is represented as x^{2}. This means multiplying the number ‘x’ by itself or ‘x’ two times. For example, 2^{2 }= 2 × 2 = 4. Similarly, 3^{2 }= 3 × 3 = 9 and so on.

### Cube

Similar to ‘squares’, the ‘cube‘ of a number ‘x’ is the number ‘x’ multiplied by itself and again multiplied by itself. It is represented as x^{3}. For example, 2^{3 }= 2 × 2 × 2 = 8, 3^{3 }= 3 × 3 × 3 = 27 and so on. Having learnt the concept now we will look at some rules of the exponential formula to learn their applications.

## Application of Exponential Formula

### Multiplication Rule

For a number ‘a’,

**a ^{m} × a^{n} = a^{m+n}**

For example, 2^{3} × 2^{2} = 2^{3 + 2} = 2^{5}. Mind you that, the base remains the same and only the powers change.

### Division Rule

For a number ‘a’,

**a ^{x} ÷ a^{y} = a^{x-y}**

For example, 2^{5 }÷ 2^{2} = 2^{5 – 2} = 2^{3}

### Power of a Power Rule

For a number ‘a’,

**\( a^{x^y} \) = a ^{x.y}**

For example, (3^{2})^{3} = 3^{2×3} = 3^{6}

### Zero Exponent

For any number ‘a’,

**a ^{0} = 1**

For example, (7^{0}) = (5^{0}) = (3^{0}) = 1

### Power of a Product Rule

For numbers ‘a’ and ‘b’,

**(ab) ^{x} = a^{x}.b^{x}**

For example, (2.3)^{5}= 2^{5}.3^{5}

### Power of a Fraction Rule

For a fraction ‘a/b’,

**\( {\left( \frac{a}{b} \right)}^x = \frac{a^x}{b^x} \) **

For example, (2/3)^{2} = 2^{2 }/ 3^{2}

### Negative Exponent

For a number ‘a’,

**a ^{-x} = \( \frac{1}{a^x} \)**

For example: 2^{-3} = 1/2^{3} = 1/8

### Fractional Exponent

For a number ‘a’,

**a ^{x/y} = ^{y}√a^{x}**

For example: 2^{1/3}=³√2

## Solved Examples for You

**Question 1: Solve the following: (-3) ^{2 }× (5/3)^{3 }**

**Answer :** (-3)^{2 }× (5/3)^{3}

= (-3 × -3) × ( ( 5 × 5 × 5 ) / ( 3 × 3 × 3 ) )

= 9 × (125/27)

= (125/3)

**Question 2: What is the exponential function equation?**

**Answer:** These are the function that are in the form f(x) = bx, where b > 0 and is not equal to 1 (b ≠ 1). In addition, just like other exponential expressions, b is called the base and x is called the exponent. However, an example of exponential function is the growth of bacteria.

**Question 3: State some real-life examples of exponential functions?**

**Answer: **Some of the common examples of exponential functions are radioactive decay, population growth, loan interest rates, etc. are the naturally occurring exponential relationships. Furthermore, this helps to predict behavior, calculate half-life or plan your budget.

**Question 4: What is the exponential relationship?**

**Answer:** These are the relationships where one of the variables is an exponent. Hence in place of it being ‘2 multiplied by x’, an exponential; relationship might have ‘2 raised to the power x’. Generally, people draw a graph to get a grasp of exponential relationships.

**Question 5: What are the rules of exponential functions?**

**Answer:** Some of the basic rules that apply to exponential functions are parent exponential functions f(x) = bx always has a horizontal asymptote at y = 0, except when b = 1. Furthermore, you can’t raise a positive number to any power and get 0 or a negative number.