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)
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