# Exponents and Powers Class 8 Notes: Chapter 12

Exponents and Powers Class 8 Notes for chapter 12 given here are a great study tool to boost productivity and improve overall knowledge about the topics. In 8th standard, the concept of exponents, powers and their applications in the real world are explained clearly. This chapter help students to build a strong foundation on the concept of exponents and powers. Solved and example problems are given here for better understanding. Students can use these notes to have a thorough revision of the entire chapter and at the same be well equipped to write the exam.

## Exponents and Powers Class 8 Concepts

The topics and subtopics covered in exponents and powers class 8 are:

• Introduction
• Powers with Negative Exponents
• Laws of Exponents
• Use of exponents to express small numbers in standard form
• Comparing very large and very small numbers

The concepts are explained clearly along with the solved examples. Go through the exponents and powers concept to solve the difficult problem.

## What Are Exponents?

Exponents are numbers which indicate how many times multiplication has to be done to a number to get the desired result. An exponent is a simple but powerful tool and is basically written above the base number on the right side. Usually, an exponent or power can either be positive or negative.

For example, if we take the number 8 we need to multiply 2 three times as in 8 = 2 x 2 x 2. Here, 2 is the base number and 3 is the exponent.

### Powers with Negative Exponents

Negative power is quite similar to the positive power of an exponent. The only difference is that in negative power the value of the expression is the correlative of the value obtained with a positive number. A negative exponent also means how many times to divide by the number.

Example: 8-1 = 1 ÷ 8 = 1/8 = 0.125

## Laws of Exponents

In instances where a and b are non-zero integers while m and n are other integers, then:

• ax an = am+n

Example: 22x23 = 22+3 = 25

• am / an = am-n

Example:  23x22 = 23-2 = 2

• (am)n = amn

Example: (22)= 22×3 =26

• am×bm = (ab)m

Example: 23x3= (2 x 3)=(6)3

• am / bm = (a / b)m

Example: 23/3=(2/3)3

• a0 = 1

Example: 20 = 1

### Uses of Exponents

It is used to express small numbers in standard form.

• Large numbers can be expressed in standard form with the help of positive exponents
• Likewise, small numbers can be expressed using negative exponents
• Exponents can be used in the comparison of very small and very large numbers.

Examples of applications of exponents are:

• Distance between Earth and Sun, which is equal to 149,600,000,000 m, can be expressed in standard form using exponents.
• Distance between Moon and Earth, equal to 384, 467, 000 m (approx).
• Speed of light which is 300,000,000 m/sec.
• The thickness of a paper, which is very small in size (0.000013cm)
• The diameter of a wire whose measure is in microns (0.000002m)
• The diameter of a human hair strand
• Height of mount Everest, equal to 8848 m.

### Expressing Large and Small Numbers

1. Standard form to write very large numbers such as n00000000000……where n is any natural number

Step 1: For any given large number, count the number of digits appearing after the left-most digit.

Step 2: To express in standard form, write down the left-most digit first.

Step 3: If the original number has digits after the left-most digit other than zero then put a decimal point and write down all the digits after the decimal till zero comes. And if there are no digits other than zero, then you can skip this step.

Step 4: Put a multiplication symbol and write down the counted digits in step-1 as an exponent to base number 10.

Let us see an example: Suppose 250000000000 is a number. To express it in standard form, we can write it as 2.5 x 1011

2. Standard form to write very small numbers in the form of 0.00000000n, where n is any natural number.

Step 1: Count the number of digits after the decimal point till the right-most last digit.

Step 2: If there is only one digit at the right-most place, then right that number and put a multiplication symbol. After that, write down the calculated digits in step-1 with a negative sign as an exponent to the base 10.

Step 3: If there are more than one digit at the end of the number. Then, first, write the right-most digit followed by a decimal point and the other non-zero digits.

Step 4: Count the number of digits in step-1, subtract the number of digits appearing in step-3 after the decimal point.

Step 5: Put a multiplication symbol and write down the calculated digits in step-4 with a minus sign as an exponent to base 10.

For example: Express 0.000000000000025 in exponent form.

Solution: In standard form, the number will be written as 2.5 x 10-14.

## Exponents and Powers Class 8 Examples

Example: Find the value of

(i)3-3

(ii) 1/2-2

Solution: (i) 3-3

The multiplicative inverse of 3-3 is 1/33

So, 1/33 = 1/(3 x 3 x3) = 1/27

(ii) 1/2-2

The multiplicative inverse of 1/2-2 is 22

Hence, 1/2-2

= 22

= 2 x 2

= 4

Example 2: Simplify-

(i) (-3)2 x (-3)-2

(ii) 25 x 23

Solution: (i) (-3)2 x (-3)-2

= (-3)2 x (1/3)2

= 9 x 1/9

= 1

(ii) 25 ÷ 23

Using the law of exponent, we can write:

=25-3

=22

=2 x 2

=4

Example 3: Find the multiplicative inverse of the following:

(i)2-3

(ii) 10-4

(iii) 7-2

(iv) 31-50

Solution: (i) The multiplicative inverse of -3 is 1/23

(iI) The multiplicative inverse of 10-4 is 1/104

(iii) The multiplicative inverse of 7-2 is 1/72

(iv) The multiplicative inverse of 31-50 is 1/3150

Example 4: Express the following numbers in standard form.

(i) 0.000015 (ii) 3050000

Solution:

(i) 0.000015 = 1.5 × 10-5

(ii) 3050000 = 3.05 × 106

### Important Questions for Practice

1. Get the multiplicative inverse of the following numbers.

(i) 10-5 (ii) 5-3 (iii) 77-2 (iv) 10-1000

2. Simplify and express it in the exponential form.

(i) (–2)-3 × (–2)-4 (ii) 32 × 3-5 × 36

3.Express 6-3 as a power with the base 3.

4. There is a stack of 6 coins each of thickness 25 mm and 6 paper notes each of thickness 0.020 mm kept in a table. What is the total thickness of the stack?

5. Express the given numbers in the usual form.

(i) 3.52 × 105 (ii) 7.54 × 10-4