Laws of exponents and radicals with examples

Rules of exponents

1. Product Rule

   • a^m.a^n=a^(m+n)
     • Ex. x^2.x^3=x^5

2. Quotient Rule

   • a^m/a^n=a^(m-n)
     • Ex. x^3/x^2=x

3. Negative Exponent Rule

   • a^-n=1/a^n
     • Ex. x^-2=1/x^2

4. Zero Exponent Rule

   • a^0=1
     • Ex. 2^0=1

5. Raising a Power to a Power

   • (a^m)^n=a^(mn)
     • Ex. (x^2)^3=x^6

6. Raising a Product to a Power

   • (ab)^m=a^m b^m
     • Ex. (2.3)^2=2^2.3^2=4.9=36

7. Raising a Quotient to a Power

   • (a/b)^m=a^m/b^m
     • Ex. (2/3)^2=2^2/3^2=4/9=36

8. Product Rule for Radicals

   • √(n&a) /√(n&b) = √(n&a/b)
   • Ex. √2 .√5 = √10

9. Quotient Rule for Radicals

   • √(n&a) .√(n&b) = √(n&ab)
   • Ex. √18 /√3 = √(18/3) = √6

10. Rational Exponents

    • Exponential form of √(n&a^m )
      • when a is nonnegative, n can be any index
        • √(n&a)= a^(1/n)
        • Ex. ∛(x^11 )=x^(11/3)
      • For any nonnegative number a, and integers m and n
        • √(n&a^m )= (√(n&a))^m=a^(m/n)
      • For any nonnegative real number a
        • √(n&a^n )=(√(n&a) )^n = a^(n/n) = a