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Home >> Exponents >> Laws of Exponents >> am ÷ an = am-n >>

Exponents Law : Division of Same Base with different Powers

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

Exponents Law 2 : am ÷ an = am-n

This law explains the division of same base with different powers i.e.(am ÷ an = am-n)

This law of Exponents applies to all non-zero integers a, where m and n can be any whole number and m is greater than n.

Study following example to prove and understand this law:

Example : Solve 25 ÷ 22
Solution: This proceeds as:

25 = 2 X 2 X 2 X 2 X 2 = 32
22 = 2 X 2 = 4

And as per the given question:
25 ÷ 22 = 32 ÷ 4 = 8 ..... (Statement 1)

Now, convert 8 into exponential form and we get:
8 = 2 X 2 X 2 = 23 .....(statement 2)

From statement 1 and 2, we get:
25 ÷ 22 = 23 ..... (Statement 3)

Now, observe carefully that base of exponential forms on both side of the above equation is same i.e. 2
And on subtracting the exponents on the L.H.S. i.e. (5 - 2), we get 3; which is equal to the exponent on the R.H.S & we get:
25 - 2 = 23 .....(Statement 4)

So from statement 3 and 4, we get:
25 ÷ 22 = 25 - 2

Hence it's proved that am ÷ an = am-n


Alternative method

25 ÷ 22 = (2 X 2 X 2 X 2 X 2) ÷ (2 X 2)

Open brackets and solve division expression & we get:
25 ÷ 22 = 2 X 2 X 2

Convert RHS into exponential form and we get:
25 ÷ 22 = 23

Since, 3 = 5 - 2; so we apply the same on the exponent of RHS and we get:
25 ÷ 22 = 25 - 2

Hence it's proved that (am ÷ an = am-n)




Explaining with the help of algebraic numbers:

Example: Prove a5 ÷ a3 = a5 - 3
Solution: Given algebraic equation is:

a5 ÷ a3 = a5 - 3

First Solve LHS and we get:
a5 ÷ a3

Expand the algebraic numbers and we get:
= (a x a x a x a x a) ÷ (a x a x a)

Open brackets and solve division expression & we get:
= a x a

Convert them to exponential form and we get:
= a2 (because variable a is repeated twice) .....(Statement 1)

Now, Solve RHS and we get:
a5 - 3

Solve (5 - 3) in exponent, as we do for whole numbers and we get:
= a2 .....(Statement 2)


From Statement 1 and 2, we get:
a2 = a2

Since, LHS = RHS = a2

Therefore it's proved that
a5 ÷ a3 = a5 - 3


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