We have already discussed areas where exponents and powers are used in real life. Now, let us understand the basic concept, before seeing how this application works. The concepts of class 8 exponents and powers are broken down into the following subtopics.

Definitions

Here are the basic definitions of the terms we use when studying powers and exponents.

• Power: When a number is multiplied by itself, it is called power expression. For example, if we multiply 5 to itself 4 times, it can be represented as follows.

5 X 5 X 5 X 5 = 5⁴. The expression 5⁴ is read as “5 to the power of 4”.

• Base: The number multiplied by itself is called the base. Thus, in the above example, 5 is the base.

• Exponent / Indices: While the term exponent is used interchangeably with power, there is a slight difference between the two. Exponent refers to the number of times the base is being multiplied by itself. In the above example (54), 4 is the exponent. The exponent is also called the index.

Representation

Summarising the definitions we studied above, we arrive at the standard representation of a power equation. Let us see the representation with another example.

Power equation:  3 to the power of 5

Representation of the equation: 3 X 3 X 3 X 3 X 3 = 3⁵. Here, the base = 3 and the exponent = 5.

Being thorough with these basic definitions and representations will help you to understand all the other concepts..

Types of Exponents

We saw that when a number (called the base) is multiplied by itself a specific number of times (exponent), we get a power expression. This is true if the exponent is a positive integer. However, when solving problems of Class 8 Exponents and Powers, you may also find the following types of exponents.

• Negative Exponents – The positive exponent means a number is multiplied by itself, as many times as the exponent. A negative exponent means the number is divided by itself as many times as the exponent.

For example, 2^-3 = {(1/2)/2}/2 = (1/2) X (1/2) X (1/2) = 1/8. Thus, we can see that 2^-3 = 1/8.

• Decimal Exponents – Another common type of exponent you will find is the decimal exponent. To calculate a number with a decimal exponent, convert the decimal to its fraction form, and then use the laws of exponents, as explained earlier.

         For example, 5^3.5 can also be written as 5^7/2.

Now that you have seen how powers and exponents relate to each other, let us move on to the next concept. There are seven laws of exponents that help to easily solve all the complicated calculations that one encounters in real-world applications.

     1. a^m×aⁿ = a^m+n

The first law of exponents is related to the multiplication of two or more power expressions when the base is the same. This is also called Like Base. Let us understand this law with an example.

3³X3² = (3X3X3) X (3X3) = 3⁵. As we can see, 3³X3² = 3³⁺² = 3⁵.

Note: This law is applicable with the same base (3 in the above example).

      2. a^m/aⁿ = a^m-n

The following law is related to the division of power expressions with a like base. To see the calculation based on this law of exponents, let us consider the following example.

5⁶/5² = (5X5X5X5X5X5) / (5X5) = (5X5X5X5) = 5⁴. In other words, 5⁶/5² = 5^6-2 = 5⁴.

      3. (a^m)ⁿ = a^mn

The third law of exponents that you will study as part of Class 8 exponents and powers is also applicable to power expressions with like bases. Check the following example to understand this law.

(2³)² = 2³ X 2³ = (2X2X2) X (2X2X2) = 2⁶. As we can see, (2³)² = 2^3x2 = 2⁶

      4. aⁿ/bⁿ = (a/b)ⁿ

So far, we have seen the laws that relate to like (same) bases. What happens when we have expressions with different bases? The next law deals with this condition. Note that here the exponent is the same. Let us understand this with the following example.

5³/2³ = (5X5X5)/(2X2X2) = (5/2) X (5/2) X (5/2) = (5/2)³. This expression can be summarised as 5³/2³ = (5/2)³.

      5. a⁰ = 1

A very important law that you need to learn as part of the Class 8 exponents and powers chapter is this one. For any expression, where the exponent is 0, the value is always 1, irrespective of the base.

• 5⁰ = 1

• 1.5⁰ = 1

• (1/3)⁰ = 1

 

     6. a^-m = 1/a^m

You already saw this law, when you were studying the negative exponents. Let us repeat this once again, to get better clarity.

7^-3 = {(1/7)/7}/7 = (1/7) X (1/7) X (1/7) = (1/7³). Thus, 7^-3 = (1/7³).

     7. a^1/n = ⁿ√a

This is an important law to remember, when you want to calculate the nth root of a number. This is best understood with the following example.

We know that the 3^rd root of 125 is 5, i.e., ³√125 = 5. This can also be expressed as follows.

125^1/3 = ³√125 = 5

So far, you have studied the basic definitions required to understand the Class 8 exponents and powers chapter. You also saw the different laws of exponents that help simplify complex calculations. Next, let us see how these laws help to solve complicated real-life problems.

Example 1: Computing Power of devices – This power is expressed in terms of GigaHertz (GHz). A simpler way to write this is 1 billion Hz or 109 Hz.

Example 2: The distances in space – When scientists calculate the distances in space like the distance between the Earth and the Sun, it is a large value. Using 147,690,000 km can be difficult in calculations. But this is easier when written as 1.4769 X 10⁸ km. Such representation is also called scientific notation.

Example 3: Molecular Calculation – If you have to write the diameter of a single human hair, how would you prefer to write it?

1. 0.00000181 m

2. 181 X 10^-6 m

The above examples surely have given you a good idea of how important it is to understand powers and exponents. This makes calculations in your daily life simple and easy.

So far you have studied a lot of concepts and the laws of exponents. Now, let us apply these concepts to some problems. This will further help you get better clarity on the topic.

Problem 1. Solve (8)³ X (8)^-2

Solution:           (8)³ X (8)^-2

                      = (8)^3+(-2)            Using the first law of exponents.

                      = (8)^3-2

                      = (8)¹

                      = 8

Problem 2. Find x such that (17)^x+3 X (17)² = (17)⁹

Solution:        (17)^x+3 X (17)² = (17)⁹

Using the first law of exponents, we can rewrite this expression as

                          (17)^x+3+2 = (17)⁹

To ensure that the left-hand side is equal to the right-hand side of the above equation, we get the following expression.

                                        x+3+2 = 9

                     => x+5     = 9

                     => x        = 9-5

                     => x        = 4

Problem 3. The expression for 9–5 as a power with the base 3 is

Solution:          9^-5

                                     = (3X3)^-5

                        = (3²)^-5

                        = (3)^2x(-5)                         Using the third law of exponents.

                        = (3)^-10

We have covered the most important concepts of Class 8 exponents and powers with the help of solved examples. You also saw the real-world applications of these concepts. If you want to understand these concepts more comprehensively, you can do so with the help of visual illustrations and further examples on the MSVGo app.