Lesson Properties and Operations of Radicals

The properties of radicals help in solving and simplifying radicals.

Adding two radicals (simplest form)
For adding two radicals, the number under the radical sign should be same in the two radicals (in the simplest form). The operation is:
a*root(n,x) + b*root(n,x) = (a+b)*root(n,x)

a*root(n,x) + b*root(n,x) = (a+b)*root(n,x)

Subtracting two radicals (simplest form)
For subtracting two radicals, the number under the radical sign should be same in the two radicals (in the simplest form). The operation is:
a*root(n,x) - b*root(n,x) = (a-b)*root(n,x)

Operations on radicals with same 'n'

root(n,x*y) = root(n,x)*root(n,y)

root(n,x*y*z) = root(n,x)*root(n,y)*root(n,z)

Operations on radicals with different 'n'

root(n,x^m) = (root(n,x))^m = (x^(1/n))^m = x^(m/n)

root(n,x)*root(m,x) = x^(1/n)*x^(1/m) = x^(1/n+1/m)

root(n,x^a)*root(m,x^b) = x^(a/n)*x^(b/m) = x^(a/n+b/m)

Other Operations
1. If m > n in root(n,x^m)

, then

root(n,x^m) = root(n,x^n*x^(m-n)) = root(n,x^n)*root(n,x^(m-n)) = x*root(n,x^(m-n))

(sqrt(x)+sqrt(y))*(sqrt(x)-sqrt(y)) = sqrt(x)^2 - sqrt(y)^2 = x - y

1/(sqrt(x)+sqrt(y)) = (sqrt(x)-sqrt(y))/((sqrt(x)+sqrt(y))*(sqrt(x)-sqrt(y))) = (sqrt(x)-sqrt(y))/(x-y)

x/sqrt(y)= x*sqrt(y)/sqrt(y)^2 = x*sqrt(y)/y