What is the square root of 10p^3

Factor 10 into its prime factors

          10 = 2 • 5  

To simplify a square root, we extract factors which are squares, i.e., factors that are raised to an even exponent.

In our case however, all the factors are only raised to the first power and this means that the square root can not be simplified

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

        sqrt (10p3)  

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(p3) = p • SQRT(p)  

Combine both simplifications

        sqrt (10p3) =

       p • sqrt(10p)  

Simplified Root :

p • sqrt(10p)

the volume is the product of the area of the base, 3x(3x+4), and the height, 3x-1

step-by-step explanation: