Laws of Exponents (Exponential Laws)

Exponents (or indices) represent repeated multiplication of a number. The laws of exponents help simplify and manipulate expressions involving powers. These laws hold for all non-zero real numbers.

1. Product Law of Exponents

For any non-zero integer $a$ and any integers m and n

aᵐ×aⁿ= aᵐ+ⁿ

This rule states that when multiplying numbers with the same base, we add the exponents.

2. Quotient Law of Exponents

For any non-zero integer $a$ and any integers $m$ and $n$:

aᵐ aⁿ=aᵐ-ⁿ ,where m>n

This rule states that when dividing numbers with the same base, we subtract the exponents.

3. Power of a Power Law

For any non-zero integer a and integers $m$ and $n$:

(aᵐ)ⁿ=aᵐ×ⁿ

This rule states that when raising a power to another power, multiply the exponents.

4. Power of a  Product Law

For any non-zero integers a and b and integer m:

(a×b)ᵐ =aᵐ ×bⁿ

This rule states that when raising a product to a power, distribute the exponent to both factors.

5. Power of a Quotient Law

For any non-zero integers a and b and integer m:

(ab)ᵐ=aᵐ bⁿ,where b≠0

This rule states that when raising a fraction to a power, distribute the exponent to both the numerator and the denominator.

6. Negative Exponent Law

For any non-zero integer a and integer m

a⁻ᵐ =1a​

A negative exponent represents the reciprocal of the positive exponent.

7. Zero Exponent Law

For any non-zero integer a

a⁰=1

Any non-zero number raised to the power of zero is always equal to 1.

 Exponents and Special Cases

1. Exponent Rule for Any Number a with Zero Exponent

   • Any non-zero number raised to the power of zero is always
   • 1. a⁰=1

2. Exponent Rule for $a=1$

   • When the base is 1, the exponent does not affect the value. 1ⁿ =1
   • for all integer values

3. Exponent Rule for a=−1

   • If the base is -1, the value depends on whether the exponent is even or odd:

     • If n is even, (-1) ⁿ = 1
     • If n is odd, (−1)ⁿ =−1

4. Infinity in Exponents

   • When the base is 1, it remains 1 for infinitely many values of n.
   • When the base is -1, it alternates between 1 and -1 based on even or odd exponents.