Class 8 Maths Notes for Chapter 12 - Exponents and Powers

Exponents and Powers

We know that the mass of the celestial bodies is huge. The mass of Earth is 5.97 x 1024 kg. If we observe 1024, 10 is the base and 24 is the exponent, that is, 10 is multiplied with itself 24 times.

Now, let’s discuss about very small numbers. For instance, the mass of subatomic particles is extremely less. The mass of a proton is 1.67 x 10–27 kg.

If we observe 10–27, 10 is the base, and –27 is the exponent. Here, the negative exponent means that 1 is divided 27 times by 10.

So, we have seen how exponents are used to represent extremely large and extremely small numbers.
Now, we will learn laws of exponents.

Laws of Exponents

Let us solve some examples using laws of exponents.

Simplify and write the answer in exponent form.

(a) (55 × 52)5 × 85

Using formula am × an = am + n in the given expression,

(55 × 52)5 × 85 = (55 + 2)5 × 85
(55 × 52)5 × 85 = (57)5 × 85
(55 × 52)5 × 85 = (57)5 × (23)5

Using formula (am)n = am × n in the above equation,

(55 × 52)5 × 85 = 57 × 5 × 23 × 5
(55 × 52)5 × 85 = 535 × 215

(b) (2/3)9 × (3/2)9 Using formula am × bm = (a × b)m in the given expression,


Now, let us read about standard form of numbers.

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NCERT Solutions for class 8 Maths Chapter 12
NCERT Solutions for class 8 Maths Chapter 12 Exercise 12.1
NCERT Solutions for class 8 Maths Chapter 12 Exercise 12.2

Standard Form

We deal with numbers every day, sometimes to measure the number of books, the height of a building, our wealth, etc. Some numbers and values come out to be very large, while some are very small. For example-

• The distance of the Sun from the nearest star (Proxima Centauri) is approximately 39,900,000,000,000 km.
• The Universe contains 50,000,000,000,000,000,000,000 stars.
• A single red blood cell is about 0.000007 meters in diameter.
• The length of the shortest wavelength of visible light (violet) is 0.0000004.

We can categorize them as-

Small Numbers - 0.000007; 0.0000004
Large Numbers - 39,900,000,000,000; 50,000,000,000,000,000,000,000

The question is, how can we write them to make our calculations easier?

We can express them in their standard form to save us from the trouble of counting the zeros or decimal places. Let’s understand the concept in detail.

Standard Form of Large Numbers

To express any large whole number in the standard form, we follow the following steps:

Step 1: Rewrite the whole number in decimal form by putting a decimal point and a zero at its end.
Step 2: Move the decimal point towards the left and place it after the first non-zero digit.
Step 3: The number of digits by which the decimal point will move to the left in the number will be the power of 10. (Power of 10 will be multiplied with the decimal number we'll get.)

Write the final form, the decimal number multiplied with the power of 10.

Let us see an example.

Express 56000000000 in standard form.
56000000000 = 56000000000.0
= 5.6 × 1010

Let’s see how we can write extremely small values.

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Standard Form of Small Numbers

To express any small whole number in the standard form, we follow the following steps:
Step 1: Move the decimal point to the right of the first non-zero digit.

Step 2: Count the places moved and this will be the power of 10. In this case, the power will be a negative integer.

Let us see an example.

Express 0.00000076 in standard form.

Move the decimal point between 7 and 6.
Since, the decimal point moves seven digits to the right, the power of 10 will be –7.
Hence, the standard form will be
0.00000076 = 7.6 x 10–7

Comparison

In this section, we will understand how to compare very large and very small numbers. Let us see an example.
The size of a cell A in an animal is 0.0000056 m and the size of cell B is 0.000000622 m. What is the size of cell A with respect to cell B?

First, we will write both the numbers in the standard form.

A = 0.0000056 = 5.6 × 10–6 ; B = 0.000000622 = 6.22 × 10–7

Now, we will compare the size of both the cells.

Hence, the size of cell A is 9 times that of cell B.

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