Question

Simplifying radicals id="TexFormula1" title=" \sqrt{20m {}^{4} } n {}^{3} " alt=" \sqrt{20m {}^{4} } n {}^{3} " align="absmiddle" class="latex-formula">
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Answer 1

Answer: 2m^2n.\sqrt{5}  

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Step-by-step explanation: Factor 20 into its prime factors

          20 = 22 • 5  

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 :

          4 = 22  

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 :

          2 = 2  

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

        2 • sqrt (5m4n3)  

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(m4n3) = m2n • SQRT(n)

sqrt (20m4n3) =

       2 m2n • sqrt(5n)  

2 m2n • sqrt(5n)