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Home >> Exponents >> Laws of Exponents >> ( am )n = am n >>

Exponents Law : One Base with Power of 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

Exponents Law 3 : (am )n = am n

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

Exponent Law am X an = am+n

This law explains how to solve one base having Power of Power i.e. [(am )n = am 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 (am X an = am+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 = 23 X 2

Hence proved [(am )n = am n ]




Explaining with the help of algebraic numbers:

Example: Prove (a2)3 = a2 x 3
Solution: Given algebraic equation is:

(a2)3 = a2 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 (am X an = am+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:
a2 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 = a2 x 3


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