NCERT Book Class 7 Maths Exponents and Powers

Exponents and Powers

13.1 INTRODUCTION

Do you know what the mass of earth is? It is

5,970,000,000,000,000,000,000,000 kg!

Can you read this number?

Mass of Uranus is 86,800,000,000,000,000,000,000,000 kg.

Which has greater mass, Earth or Uranus?

Distance between Sun and Saturn is 1,433,500,000,000 m and distance between Saturn and Uranus is 1,439,000,000,000 m. Can you read these numbers? Which distance is less?

These very large numbers are difficult to read, understand and compare. To make these numbers easy to read, understand and compare, we use exponents. In this Chapter, we shall learn about exponents and also learn how to use them.

13.2 EXPONENTS

We can write large numbers in a shorter form using exponents.

Observe 10, 000 = 10 × 10 × 10 × 10 = 104

The short notation 104 stands for the product 10×10×10×10. Here ‘10’ is called the base and ‘4’ the exponent. The number 104 is read as 10 raised to the power of 4 or simply as fourth power of 10. 10⁴ is called the exponential form of 10,000.

We can similarly express 1,000 as a power of 10. Since 1,000 is 10 multiplied by itself three times,

                                                  1000 = 10 × 10 × 10 = 10³

 We have used numbers like 10, 100, 1000 etc., while writing numbers in an expanded This can be written as 4 × 104 + 7 ×103 + 5 × 102 + 6 × 10 + 1. Try writing these numbers in the same way 172, 5642, 6374.

In all the above given examples, we have seen numbers whose base is 10. However the base can be any other number also. For example:

81 = 3 × 3 × 3 × 3 can be written as 81 = 34, here 3 is the base and 4 is the exponent.

Some powers have special names. For example,

102, which is 10 raised to the power 2, also read as ‘10 squared’ and 103, which is 10 raised to the power 3, also read as ‘10 cubed’. Can you tell what 53 (5 cubed) means?

                                                5³ = 5 × 5 × 5 = 125

So, we can say 125 is the third power of 5. What is the exponent and the base in 53?

Similarly, 2⁵ = 2 × 2 × 2 × 2 × 2 = 32, which is the fifth power of 2. In 25, 2 is the base and 5 is the exponent.

In the same way,                      243 = 3 × 3 × 3 × 3 × 3 = 35

                                                 64 = 2 × 2 × 2 × 2 × 2 × 2 = 26

                                                  625 = 5 × 5 × 5 × 5 = 5⁴

Find five more such examples, where a number is expressed in exponential form. Also identify the base and the exponent in each case. You can also extend this way of writing when the base is a negative integer. What does (–2)3 mean?

                                                  It is (–2)³ = (–2) × (–2) × (–2) = – 8

                                                  Is (–2)⁴ = 16? Check it.

Instead of taking a fixed number let us take any integer a as the base, and write the numbers as,

                                                    a × a = a² (read as ‘a squared’ or ‘a raised to the power 2’)

                                                    a × a × a = a³ (read as ‘a cubed’ or ‘a raised to the power 3’)

                                                    a × a × a × a = a⁴ (read as a raised to the power 4 or the 4th power of a)
                                                     ..............................

                                                    a × a × a × a × a × a × a = a⁷ (read as a raised to the power 7 or the 7^th power of a) and so on.

                                                    a × a × a × b × b can be expressed as a³b² (read as a cubed b squared)

Please refer to attached file for NCERT Class 7 Maths Exponents and Powers