Exponents Law : Division of Same Base with different Powers at Algebra Den

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Home >> Exponents >> Laws of Exponents >> a^m ÷ aⁿ = a^m-n >>

Exponents Law : Division of Same Base with different Powers

a^m X aⁿ = a^m+n a^m ÷ aⁿ = a^m-n ( a^m )ⁿ = a^{m n} a^m X b^m = (ab)^m a^m ÷ b^m = (a/b)^m
a⁰ = 1 a^-m

Exponents Law 2 : a^m ÷ aⁿ = a^m-n

This law explains the division of same base with different powers i.e.(a^m ÷ aⁿ = a^m-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:
2^{5 - 2} = 2³ .....(Statement 4)

So from statement 3 and 4, we get:
2⁵ ÷ 2² = 2^{5 - 2}

Hence it's proved that a^m ÷ aⁿ = a^m-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 = 2^{5 - 2}

Hence it's proved that (a^m ÷ aⁿ = a^m-n)

Explaining with the help of algebraic numbers:

Example: Prove a⁵ ÷ a³ = a^{5 - 3}
Solution: Given algebraic equation is:

a⁵ ÷ a³ = a^{5 - 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:
a^{5 - 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 = a^{5 - 3}