Exponents
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) 1^0,7 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
Which number should we use next to form the pattern?
Good, so 10000000 can also be written as
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
Exponential Notation: 10,000 =
In this notation,
components of the Exponential Expression:
Base:
The '10' in
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
The exponent indicates how many times the base is multiplied by itself.
Here, '4' means that '10' is multiplied four times.
Reading and Interpretation
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,
We can similarly express 1,000 as a power of 10.
Exponents with Bases Other Than 10:
Can you tell what
So, we can say 125 is the third power of 5. What is the exponent and the base in
Exponent :
Negative Integer Bases in Exponential Forms:
You can also extend this way of writing when the base is a negative integer.What does
Similarly,
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 =
a × a × a =
(i)Express 256 as a power 2.
- We have 256 =
- Expressing it with the help of prime factorisation
- We have found the answer
(ii)Which one is greater
=8 2 - Meanwhile,
=2 8 - We have found the respective values which tells us that:
2 8 8 2 - We have found the answer
Expand
a 3 isb 2 times a multiplied bytimes b - Thus we get
- Similarly,
a 2 =b 3 times a xtimes b b 2 =a 3 times b xtimes a b 3 =a 2 times b xtimes a - This tells us that
the given exponent products are equal to each other. - However,
a 3 =b 2 b 2 anda 3 b 3 =a 2 a 2 b 3 - We have found the answer.
Note that in the case of terms
Thus
On the other hand,
Thus,
EXAMPLE
Express the following numbers as a product of powers of prime factors:
(i) 72
72 | ||||||||||
2 | × | 36 | ||||||||
2 | × | 18 | ||||||||
2 | × | 9 | ||||||||
3 | × | 3 | ||||||||
72 | = | 2 | × | 2 | × | 2 | × | 3 | × | 3 |
Thus, 72 = 2
(ii)Find the remaining numbers as a product of powers of prime factors (ii) 432 (iii) 1000 (iv) 16000
Let's solve this example in another way:
1000 = 10 × 100 = 10 × 10 × 10
= (2 × 5) × (2 × 5) × (2 × 5) (Since 10 = 2 × 5)
= 2 × 5 × 2 × 5 × 2 × 5 = 2 × 2 × 2 × 5 × 5 × 5
or 1000 = 2
Is this method correct?
EXAMPLE 4
Work out