This topic gives an overview of;
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 × 10^{24} kg. We read 10^{24} as 10 raised to the power 24. We know 2^{5 }= 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?
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
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
Find the value of
Simplify
(i)
Express as a power with the base 2.
We have,
Simplify and write the answer in the exponential form.
On both the sides powers have the same base different from 1 and – 1, so their exponents must be equal
Observe the following facts.
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 × 10^{11}
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.
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 × 10^{24} kg and mass of moon is 7.35 × 10^{22} kg. What is the total mass?
Total mass = 5.97 × 10^{24} kg + 7.35 × 10^{22 }kg.
= 5.97 × 100 × 10^{22} + 7.35 × 10^{22}
= 597 × 10^{22} + 7.35 × 10^{22}
= (597 + 7.35) × 10^{22}
= 604.35 × 10^{22} kg.
The distance between Sun and Earth is 1.496 × 10^{11}m and the distance between Earth and Moon is 3.84 × 10^{8}m.
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
Express the following numbers in standard form.
(i) 0.000035 (ii) 4050000
Solution
(i0.000035 = 3.5 × 10^{– 5}(ii) 4050000 = 4.05 × 10^{6}
Express the following numbers in usual form.
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.
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