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Home >> Exponents >> Laws of Exponents >> am ÷ bm = (a/b)m >>

Exponents Law : Division of different Bases with Same Power

am X an = am+n am ÷ an = am-n ( am )n = am n am X bm = (ab)m am ÷ bm = (a/b)m
a0 = 1 a-m

Exponent Law 5 : am ÷ bm = (a/b)m

This law explains division of different Bases with Same Exponents am ÷ bm = (a/b)m
This law of Exponents applies to all non-zero integers a & b, where m can be any whole number

Study following example to prove and understand this law:

Example: Solve 23 ÷ 33
Solution: This proceeds as:

23 ÷ 33

Expand the given exponential forms and we get:
= (2 X 2 X 2) ÷ (3 X 3 X 3)

Or we can also write it as:
= 2 ÷ 3 X 2 ÷ 3 X 2 ÷ 3

Now, observe carefully you can prepare three groups i.e.:
= (2 ÷ 3) X (2 ÷ 3) X (2 ÷ 3)

And this can also be written into following exponential form:
= (2 ÷ 3)3 [because ((2 ÷ 3) is written thrice]

Hence, we get
23 ÷ 33 = (2 ÷ 3)3
And this explained Law of exponent i.e. am ÷ bm = (a/b)m



Explaining with the help of algebraic numbers

Prove: a5 ÷ b5 = (a ÷ b)5
Solution: Given algebraic expression is:

a5 ÷ b5 = (a ÷ b)5

Let's first solve the LHS:
a5 ÷ b5

Expand the given exponential forms and we get:
= (a x a x a x a x a) ÷ (b x b x b x b x b)

Or we can also write it as:
= a ÷ b x a ÷ b x a ÷ b x a ÷ b x a ÷ b

Now, observe carefully you can prepare groups as shown below:
= (a ÷ b) x (a ÷ b) x (a ÷ b) x (a ÷ b) x (a ÷ b) ..... (Statement 1)

Now, solve the RHS:
(a ÷ b)5

Expand the given exponential form and we get:
= (a ÷ b) x (a ÷ b) x (a ÷ b) x (a ÷ b) x (a ÷ b) ..... (Statement 2)

From Statement 1 and 2, we get:
LHS = RHS

Hence it's proved that:
a5 ÷ b5 = (a ÷ b)5

And this explained Law of exponent i.e. am ÷ bm = (a/b)m

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