Simplify the square root of 18m^9

3 m4 • sqrt (2m)

Step-by-step explanation:

STEP 1: Factor 18 into its prime factors

18 = 2 • 32

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:

9 = 32

Factors which will remain inside the root are:

2 = 2

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:

3 = 3

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

3 • sqrt (2m9)

STEP 2: 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 (m9) = m4 • SQRT (m)