Laws of Exponents
The laws of exponents, also known as the rules of indices, are mathematical rules that describe how to handle operations involving exponents. They are fundamental in algebra and simplify the process of working with large numbers and variables.
Here are the key laws of exponents:
Multiplying Powers with the Same Base
(i) Let us calculate
- We have
2 2 - and
2 3 - If we take the given product
- Together we get
- calculating the powers
- Hence we get the value of 2^5
Note that the base in
(ii)
- We have
− 3 4 - We have
− 3 3 - If we take the given product
- Together we get
- calculating the powers
- Hence we get the value of 3^7
Again, note that the base is same and the sum of exponents, i.e. 4 + 3 is 7
Caution! Consider
Can you add the exponents?
Do you see ‘why’? The base of
(i) Let us simplify
- We have
3 7 ÷ 3 4 - which means
- we can cancel out the common factors
- calculating the powers we get:
- Hence we have found the value of
3 7 ÷ 3 4
Note, in
(ii) Let us simplify
- We have
5 6 5 2 - which means
- we can cancel out the common factors
- calculating the powers we get
- Hence we have found the value of
5 6 5 2
Consider the following:
Simplify
2 3 2 2 3 times with itself. - If we multiply
2 3 × 2 3 - This is also equal to
2 6 - Which can be written as
- Thus, we have proved.
From this we can generalise for any non-zero integer ‘a’, where ‘m’ and ‘n’ are whole numbers,
Can you tell which one is greater
5 2 × 3 means5 2 by 3 - which is equal to
- Meanwhile,
5 2 3 5 2 times - Which gives us
- Comparing
5 2 3 5 2 × 3 - Therefore, we get the result
(i)Let us simplify
Notice that here the two terms
- Now,
2 3 × 3 3 - We note that this can also be expressed as
- Observe 6 is the
of bases 2 and 3 - This can further be written as 2x2x2x3x3x3 or 2x3 x 2x3 x 2x3
- Thus, we can say that
2 3 x 3 3 6 3
(ii)Let us simplify
- Now,
4 4 × 3 4 - We note that this can also be expressed as
- Observe 12 is the
of bases 3 and 4 - This can further be written as 4x4x4x4x3x3x3x3 or 4x3 x 4x3 x 2x3 x 4x3
- Thus, we can say that
4 4 × 3 4 12 4
In general, for any non-zero integer a
Express the following terms in the exponential form:
(i)
- Now,
2 × 3 5 - Can also be written as
- This gives us
- Thus, we can write
Observe the following simplifications:
From these examples we may generalise
Can you tell what
By using laws of exponents.
So
Can you tell what
And
Therefore
Thus
So, we can say that any number (except 0) raised to the power (or exponent) 0 is 1.
Example
Let us solve some examples using rules of exponents developed.
Write exponential form for 8 × 8 × 8 × 8 taking base as 2.
- We know that 8 =
when base taken as 2 - We can re-write as:
- Using
a m n = a mn 8 4 - Therefore we have found the answer
Simplify and write the answer in the exponential form.
Match the below exponents
Note: In most of the examples that we have taken in this chapter, the base of a power was taken an integer. But all the results of the chapter apply equally well to a base which is a rational number.