Factor 48 into its prime factors
48 = 24 • 3
To simplify a square root, we extract factors which are squares, i.e., factors that are raised to an even exponent.
Factors which will be extracted are :
16 = 24
Factors which will remain inside the root are :
3 = 3
To complete this part of the simplification we take the squre root of the factors which are to be extracted. We do this by dividing their exponents by 2 :
4 = 22
At the end of this step the partly simplified SQRT looks like this:
4 • sqrt (3x3y4)
Step 2 :Simplify the Variable part of the SQRT
Rules for simplifing variables which may be raised to a power:
(1) variables with no exponent stay inside the radical
(2) variables raised to power 1 or (-1) stay inside the radical
(3) variables raised to an even exponent: Half the exponent taken out, nothing remains inside the radical. examples:
(3.1) sqrt(x8)=x4
(3.2) sqrt(x-6)=x-3
(4) variables raised to an odd exponent which is >2 or <(-2) , examples:
(4.1) sqrt(x5)=x2•sqrt(x)
(4.2) sqrt(x-7)=x-3•sqrt(x-1)
Applying these rules to our case we find out that
SQRT(x3y4) = xy2 • SQRT(x)
Combine both simplifications
sqrt (48x3y4) =
4 xy2 • sqrt(3x)
Simplified Root :
4 xy2 • sqrt(3x)
Hope this helps~!
Happy studying~!
~{Dunsforhands}