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Home >> Exponents >> Laws of Exponents >> a0 = 1 >>

Exponents Law : any non-zero integer having exponent as Zero is always equals to 1

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 6 : a0 = 1

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

Law of Exponent : am ÷ an = am-n

This law explains that any non-zero integer having exponent as Zero is always equals to 1

Algebraic representing of this law is
a0 = 1

Let proved this with the help of following example:

Example: Solve a5 ÷ a5
Solution: Given exponential forms:
a5 ÷ a5

Expand the given exponential form and we get
= a x a x a x a x a ÷ a x a x a x a x a

Solve the division expression and we get:
= 1

Therefore we get:
a5 ÷ a5 = 1 ..... (Statement 1)

Now once again take the given exponential forms:
a5 ÷ a5

Apply the law of exponential i.e. am ÷ an = am-n and we get:
= a5 - 5

Solve the exponents and we get:
= a0

Therefore we get:
a5 ÷ a5 = a0 ..... (Statement 2)


From Statement 1 and 2, we get:
a0 = 1

Hence, this proves the law:
a0 = 1



Study the following example to further understand this:

Example: Solve 23 ÷ 23
Solution: Given exponential forms:
23 ÷ 23

Expand the given exponential form and we get
= 2 X 2 X 2 ÷ 2 X 2 X 2

Solve the division expression and we get:
= 1

Therefore we get:
23 ÷ 23 = 1 ..... (Statement 1)

Now once again take the given exponential forms:
23 ÷ 23

Apply the law of exponential i.e. am ÷ an = am-n and we get:
= 23 - 3

Solve the exponents and we get:
= 20
Therefore we get:
23 ÷ 23 = 20 ..... (Statement 2)


From Statement 1 and 2, we get:
20 = 1
Hence, this proves the law:
a0 = 1

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