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Powers and Exponents

    Introduction

    Exponential Form

    Writing a number in exponential form through prime factorization

    Laws of Exponents

    Exercise 11.1

    Exercise 11.2

    Extra Curriculum Support

    Looking Back

    Easy Level Worksheet

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Glossary

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Powers and Exponents > Exponential Form

Exponential Form

Consider the number 10000000.

Let us say we are lazy people. I don't want to write such a big number. Can I come up with a short hand notation?

Lets play around and create some short hand notations. I can think of the following:

1[0,7] (or) 10,7 etc.

We can come up with any number of combinations like this. But we are not applying any mathematical knowledge in coming up with a representation. We are mathematicians and we can do much better.

The number is 10000000.

This can also be written as 100·100·100·10. There seems to be some pattern. But it's not perfect.

Which number should we use next to form the pattern?

Good, so 10000000 can also be written as 10·10·10·10·10·10·10. Or 10 multiplied 7 times. Now there is a pattern.

An exponent tells you how many times to multiply a number by itself.

The concept of using exponents allows us to write large numbers more compactly.Let's see an example:

The Number 10,000 and Its Exponential Form:

Observation: 10,000 = 10 × 10 × 10 × 10
Here, the number 10,000 is being expressed as the product of the number 10 multiplied by itself times.

Exponential Notation: 10,000 = 104

In this notation,
104 is a compact way of writing the 10,000. Now, let's understand the
components of the Exponential Expression: 104

Base:
The '10' in 104 is called the base.
The base is the number that is being repeatedly multiplied.
In this case, it means we are multiplying '10' several times.

Exponent:
The '4' in 104 is called the exponent or power.
The exponent indicates how many times the base is multiplied by itself.
Here, '4' means that '10' is multiplied four times.

Reading and Interpretation

104 is read as "10 raised to the power of 4" or "the fourth power of 10".
This exponential form is a concise way of representing larger numbers. Exponents make it easier to write and compute with very large or very small numbers.

Summary
In short,104 efficiently expresses the number , with '10' as the base (the number being multiplied) and '4' as the exponent (indicating the number of times '10' is multiplied by itself). This method of using exponents is widely used in mathematics, science, and engineering to simplify the representation of large numbers.

We can similarly express 1,000 as a power of 10.

1000 = 10 × 10 × 10 = 10
Here again, 103 is the exponential form of 1,000.
Similarly, 1,00,000 = 10 × 10 × 10 × 10 × 10 = 10
So, 105 is the exponential form of 1,00,000
In both these examples, the base is 10; in case of 103, the exponent is and in case of 105 the exponent is .
In all the above given examples, we have seen numbers whose base is 10. However the base can be any other number also.
81 = 3 × 3 × 3 × 3 can be written as 81 = 34, here is the base and is the exponent.

Find five more such examples, where a number is expressed in exponential form. Also identify the base and the exponent in each case.

(i)34 = 3 × 3 × 3 × 3 =

In 34, 3 is Known as and 4 as an .

(ii)44 = 4 × 4 × 4 × 4 =

In 44, 4 is Known as and 4 as an .

(iii)75 = 7 × 7 × 7 × 7 × 7=

In 75, 7 is Known as and 5 as an .

(iv)53 = 5 × 5 × 5 =

In 53, 5 is Known as and 3 as an .

(v)26 = 2 × 2 × 2 × 2 × 2 × 2=

In 26, 2 is Known as and 6 as an .

Exponents with Bases Other Than 10:

Can you tell what 53 (5 cubed) means?

53 = 5 × 5 × 5 =

So, we can say 125 is the third power of 5. What is the exponent and the base in 53? Exponent : , Base :

Negative Integer Bases in Exponential Forms:

You can also extend this way of writing when the base is a negative integer.What does −23 mean?

−23 = (–2) × (–2) × (–2) =

Similarly, −24 =

Special Names for Powers (e.g., Squared, Cubed):

Instead of taking a fixed number let us take any integer a as the base, and write the numbers as,

a × a = a2 (read as 'a squared' or 'a raised to the power 2') {.reveal(when="blank-4")}a × a = a2 (read as 'a squared' or 'a raised to the power 2')

a × a × a = a3 (read as 'a cubed' or 'a raised to the power 3') {.reveal(when="blank-4")}a × a × a = a3 (read as 'a cubed' or 'a raised to the power 3')

Express:

(i)729 as a power of 3

729 = 3

36 = 3 × 3 × 3 × 3 × 3 × 3 =

(ii)128 as a power of 2

128 = 2

27 = 2 × 2 × 2 × 2 × 2 × 2 × 2=

(iii)343 as a power of 7

343 = 7

73 = 7 × 7 × 7 =

Express 256 as a power 2.

256 as a power 2

  • We have 256 =
  • Expressing it with the help of prime factorisation
  • We have found the answer

Which one is greater 82 or 28?

Which is greater?

  • 82=
  • Meanwhile, 28=
  • We have found the respective values which tells us that: 28 82
  • We have found the answer

Which one is greater 82 or 28?

Solution:

82 = 8 × =

28 = 2 × 2 × 2 × 2 × × × 2 × 2 =

Clearly, 28 82

Expand a3b2, a2b3, b2a3, b3a2. Are they all same?

Evaluate the given exponent products

  • a3b2 is times a multiplied by times b
  • Thus we get
  • Similarly, a2b3 = times a x times b
  • b2a3 = times b x times a
  • b3a2 = times b x times a
  • This tells us that the given exponent products are equal to each other.
  • However, a3b2 = b2a3 and b3a2 = a2b3
  • We have found the answer. :::

Note: That in the case of terms a3 b2 and a2 b3 the powers of a and b are different.

Thus a3 b2 and a2 b3 are different.

On the other hand, a3 b2 and b2 a3 are the same, since the powers of a and b in these two terms are the same. The order of factors does not matter.

Thus, a3 b2 = a3 × b2 = b2 × a3 = b2 a3. Similarly, a2 b3and b3 a2 are the .