Exponents Law : One Base with Power of Power at Algebra Den

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Home >> Exponents >> Laws of Exponents >> ( a^m )ⁿ = a^{m n} >>

Exponents Law : One Base with Power of Power

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 3 : (a^m )ⁿ = a^{m n}

Before you understand this law, you are advice to read:

Exponent Law a^m X aⁿ = a^m+n

This law explains how to solve one base having Power of Power i.e. [(a^m )ⁿ = a^{m n} ]
This law of Exponents applies to all non-zero integers a, where m and n can be any whole number

Study following example to prove and understand this law:

Example : Solve (2²)³
Solution: This proceeds as:

Given exponential form = (2²)³

This means that (2²) is multiplied thrice with itself i.e.:

(2²)³ = 2² X 2² X 2²

Apply Law of exponent (a^m X aⁿ = a^m+n ) and we get:
(2²)³ = 2²⁺²⁺²

Add exponents on RHS and we get:
(2²)³ = 2⁶

Since 6 = 3 X 2, so we apply the same on the exponent of RHS & we get
(2²)³ = 2^{3 X 2}

Hence proved [(a^m )ⁿ = a^{m n} ]

Explaining with the help of algebraic numbers:

Example: Prove (a²)³ = a^{2 x 3}
Solution: Given algebraic equation is:

(a²)³ = a^{2 x 3}

First Solve LHS and we get:
(a²)³

This means that (a²) is multiplied thrice with itself i.e.:
= a² x a² x a²

Apply Law of exponent (a^m X aⁿ = a^m+n ) and we get:
= a²⁺²⁺²

Add exponents as we do addition of whole numbers and we get:
= a⁶ ..... (Statement 1)

Now, Solve RHS and we get:
a^{2 x 3}

Multiply exponents (2 X 3), as we do for whole numbers and we get:
= a⁶ ..... (Statement 2)

From Statement 1 and 2, we get:
a⁶= a⁶

Since, LHS = RHS = a⁶

Therefore it's proved that
(a²)³ = a^{2 x 3}