Objective:

This topic gives an overview of; 

• Introduction of Exponents
• Powers with Negative Exponents
• Laws of Exponents
• Use of Exponents to Express Small Numbers in Standard Form
• Comparing very large and very small numbers

Introduction

Mass of earth is 5,970,000,000,000, 000, 000, 000, 000 kg.We have already learnt in earlier class how to write such large numbers more conveniently using exponents, as, 5.97 × 10²⁴ kg. We read 10²⁴ as 10 raised to the power 24. We know 2⁵ = 2 × 2 × 2 × 2 × 2and 2^m = 2 × 2 × 2 × 2 × ... × 2 × 2 ... (m times).

Let us now find what is 2^{– 2} is equal to?

Powers with Negative Exponents

YOu know that,

Continuing the above pattern we get,

Similarly

What is equal to ? Now consider the following:

So looking at the above pattern, we say

You can now find the value of in a similar manner.

We have,

In general, we can say that for any non-zero integer where is a positive integer. is the multiplicative inverse of

Laws of Exponents

We have learnt that for any non-zero integer where m and n are natural numbers. Does this law also hold if the exponents are negative? Let us explore.

In general, we can say that for any non-zero integer

Where and are integers

Example 1

Find the value of

Solution

Example 2

Simplify

Solution

(i)

Example 3

Express as a power with the base 2.

Solution

We have,

Example 4

Simplify and write the answer in the exponential form.

Solution

Example 5

Solution

On both the sides powers have the same base different from 1 and – 1, so their exponents must be equal

Example 6

Solution

Example 7

Soltion

Use of Exponents to Express Small Numbers in Standard Form

Observe the following facts.

1. The distance from the Earth to the Sun is 149,600,000,000 m.
2. The speed of light is 300,000,000 m/sec.
3. Thickness of Class VII Mathematics book is 20 mm.
4. The average diameter of a Red Blood Cell is 0.000007 mm.
5. The thickness of human hair is in the range of 0.005 cm to 0.01 cm.
6. The distance of moon from the Earth is 384, 467, 000 m (approx).
7. The size of a plant cell is 0.00001275 m.
8. Average radius of the Sun is 695000 km.
9. Mass of propellant in a space shuttle solid rocket booster is 503600 kg.
10. Thickness of a piece of paper is 0.0016 cm.
11. Diameter of a wire on a computer chip
12. The height of Mount Everest is 8848 m.

Observe that there are few numbers which we can read like 2 cm, 8848 m, 6,95,000 km. There are some large numbers like 150,000,000,000 m and some very small numbers like 0.000007 m.

Identify very large and very small numbers from the above facts and write them in the adjacent table:

We have learnt how to express very large numbers in standard form in the previous class.

For example: 150,000,000,000 = 1.5 × 10¹¹

Now, let us try to express 0.000007 m in standard form.

Similarly, consider the thickness of a piece of paper which is 0.0016 cm.

Comparing very large and very small numbers

The diameter of the Sun is and the diameter of the Earth is

Suppose you want to compare the diameter of the Earth, with the diameter of the Sun.

Diameter of the Sun

Diameter of the earth

So, the diameter of the Sun is about 100 times the diameter of the earth.

Let us compare the size of a Red Blood cell which is 0.000007 m to that of a plant cell which is 0.00001275 m.

Size of Red Blood cell

Size of plant cell

So a red blood cell is half of plant cell in size.

Mass of earth is 5.97 × 10²⁴ kg and mass of moon is 7.35 × 10²² kg. What is the total mass?

Total mass = 5.97 × 10²⁴ kg + 7.35 × 10²² kg.

= 5.97 × 100 × 10²² + 7.35 × 10²²

= 597 × 10²² + 7.35 × 10²²

= (597 + 7.35) × 10²²

= 604.35 × 10²² kg.

The distance between Sun and Earth is 1.496 × 10¹¹m and the distance between Earth and Moon is 3.84 × 10⁸m.

During solar eclipse moon comes in between Earth and Sun.

At that time what is the distance between Moon and Sun.

Distance between Sun and Earth

Distance between Earth and Moon

Distance between Sun and Moon

Example 8

Express the following numbers in standard form.

(i) 0.000035 (ii) 4050000

Solution

(i0.000035 = 3.5 × 10^{– 5}(ii) 4050000 = 4.05 × 10⁶

Example 9

Express the following numbers in usual form.

Solution

Summary:

• Whole numbers can be expressed in standard form, in factor form and in exponential form.

• Exponential notation makes it easier to write a number as a factor repeatedly.

• A number written in exponential form is a base raised to an exponent.

• The exponent tells us how many times the base is used as a factor.