√125x^4y^2
simplify square roots

5 x2 • sqrt(5y)

Step-by-step explanation:

Factor 125 into its prime factors

125 = 53

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 :

25 = 52

Factors which will remain inside the root are :

5 = 5

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 :

5 = 5

At the end of this step the partly simplified SQRT looks like this:

5 • sqrt (5x4y)

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(x4y) = x2 • SQRT(y)

Combine both simplifications

sqrt (125x4y) =

5 x2 • sqrt(5y)

Simplified Root :

5 x2 • sqrt(5y)