10.4. 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 695,000 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 is 0.000003 m.
12. The height of Mount Everest is 8848 m.

This segment lists various measurements of both large and small physical quantities. It highlights how some numbers are easy to handle, like 20 mm or 8848 m, while others are cumbersome, such as 149,600,000,000 m or 0.000007 m. The main goal here is to prepare the understanding that very large and very small numbers require a different method of representation, which we achieve through 'standard form'— a concise way to express these numbers using exponents.

Imagine trying to write your age in seconds. For instance, at age 10, you'd write '315,360,000 seconds.' That's a lot of digits! Instead, if we write this in standard form, it would be in a much simpler notation: approximately 3.15 × 10^8 seconds.

Now, let us try to express 0.000007 m in standard form. 0.000007 = 7 × 10–6 m. Similarly, consider the thickness of a piece of paper which is 0.0016 cm. 0.0016 = 1.6 × 10–3 cm.

To convert numbers like 0.000007 into standard form, we move the decimal point to the right until we reach the first non-zero digit. This movement indicates how many places we moved, which becomes the exponent. Thus, 0.000007 becomes 7 × 10^–6. For 0.0016, when moving the decimal point three places to the right, it becomes 1.6 × 10^–3.

Think of this process like zooming in on a number. When you're far away, 0.000007 seems tiny and hard to read, but once you zoom in, you see the prominent '7,' which tells you the real size is much easier to manage.

Diameter of the Sun = 1.4 × 10^9 m and the diameter of the Earth is 1.2756 × 10^7 m. Therefore, Diameter of the Sun ÷ Diameter of the Earth = 1.4 × 10^9 / 1.2756 × 10^7 = approximately 100.

To compare values expressed in standard form, we can divide them. Here, we take the diameter of the Sun and divide it by that of the Earth. By simplifying, we align the exponents (10^9 vs. 10^7) in the division, leading us to the conclusion that the diameter of the Sun is approximately 100 times larger than that of the Earth.

Imagine having two different-sized watermelons. If the larger watermelon is 1.4 meters around while the smaller one is 1.27 meters around, to find out how many times bigger the larger one is, you can simply compare their sizes using some basic math, showing just how significant the size difference really is.

Total mass = 5.97 × 10^24 kg + 7.35 × 10^22 kg = (597 + 7.35) × 10^22 = 604.35 × 10^22 kg.

When adding numbers in standard form, it’s crucial to have the same exponent. Here, we convert both numbers to have the same exponent of 10^22 by recognizing that 5.97 × 10^24 equals 597 × 10^22. Now, we can simply add the coefficients (597 + 7.35) to get 604.35 × 10^22.

This process is similar to adding prices at a supermarket. If you know you spent $5.97 on one item and$0.07 on another, you'd appreciate that while one cost is significantly larger, adding them together still gives you the total you need to pay without changing the overall value much.

Express the following numbers in usual form: (i) 3.52 × 10^5 = 352000, (ii) 7.54 × 10^–4 = 0.000754.

To convert from standard form (e.g., 3.52 × 10^5) to usual form, we multiply the decimal coefficient by 10 raised to the corresponding exponent. For 3.52 × 10^5, we move the decimal 5 places to the right to yield 352000. Conversely, for 7.54 × 10^–4, we move the decimal 4 places to the left, resulting in 0.000754.

Think of this like filling a jar with marbles. If you have a big bag of 100,000 marbles and you want to show how many you have in a small space, you can count it as just 100,000 or break it down to know that half of that is hidden under the couch, representing those smaller amounts in tricky but manageable figures.

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