Laws of Exponents

The laws of exponents are used for combining exponents of numbers. Exponents is a number raised to another number, it is denoted as,   a n,  here, n is known as the exponent of the nth power of a.

Laws of exponent are as follows:

x1 = x

x0 = 1

Negative exponent

x - n = (1)/(x^n)

Multiplication law of exponent

x a x b =  (x) a+b

Division law of exponent

(x^a)/(x^b)   =  (x) a-b

Power of power law of exponent

(x a) b  =  x ab

(xy)a   =  xa y a(x/y)^a = (x^a)/(y^a)

Fractional law of exponent x^(a/b)     =    (x^a)^(1/b)  =   root(b)(x^a)


Examples on laws of exponents

1)  Solve  the exponent (32) 5 =  (3) 2x5 = (3) 10      ( using Multiplication law)

2) Solve the exponent  (5^4)/(5)  =  (5) 4-1 =  5 3 = 125    (Using division law)

3) Simplify the exponent 2 (- 5)  = (1)/(2^5)  = (1)/(32)

4) Simplify the exponent  (1/4)^(-3) = (1)/[(1/4)^3] = (1)/(1^3/4^3)  = (4^3)/(1^3) = 4 3 = 64   (using Division law)

5)Simplify using the law of exponents  (sqrt(4) ) -3 = (4^(1/2))^(-3)        (using fractional law)

=  (4)^[(1/2)*(-3)]   (using power of power  law)

= 4^(-3/2)   (using multiplication law)

=  (1)/(4^(3/2))           (using negative law)

= (1)/((4^3)^(1/2)) = (1)/((64)^(1/2))

= (1)/((8^2)^(1/2)) = (1)/(8^(2*(1/2))) = (1)/(8)

Solved examples

Below are the solved examples on laws of exponents:

1)Solve the exponents  3 7 * 3 2 = 3 (7+2) = 3 9       (using Multiplication law)

2) Solve the exponents 2 (-3) * (-7) (-3) =  (2 * (-7)) (-3) =  (-14) (-3)    (using Power of power law)

3) root(3)((343)^-2)  = (343^(-2))^(1/3)      (using Fractional law)

=  (343^(1/3))^(-2) =  (1)/((343^(1/3))^2)   ( using negativel law)

=  (1)/(7^3^(1/3))^2   = (1)/(7^(3*1/3))^2       (using power of power law)

=  (1)/(7^2)  =   (1)/(49)

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4) Simplify using the law of exponents [{(1/5)^(-2)}^2]^(-1) =  {(1/5)^(-2)}^(2*(-1))

=  {(1/5)^(-2)}^(-2)

= (1/5)^((-2)*(-2))

= (1/5)^4

= (1^4)/(5^4) = (1)/(625)