Question 164461
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{{{17 sqrt(20)}}}

Break {{{20}}} down into a product of prime numbers:

{{{20=2*2*5}}}

{{{17sqrt(2*2*5)}}}

Group the pair of like factors:

 {{{17sqrt((2*2)*5)}}}

The pair of like factors {{{2*2}}} can be written as 
a square {{{2^2}}}

 {{{17sqrt((2^2)*5)}}}

Take individual square roots:

{{{17sqrt(2^2)sqrt(5)}}}

The square root of the square of 2 gives only 2:

{{{17*2sqrt(5)}}}

{{{34sqrt(5)}}}

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{{{-7 sqrt(242)}}}

Break {{{242}}} down into a product of prime numbers:

{{{20=2*11*11}}}

{{{-7sqrt(2*11*11)}}}

Group the pair of like factors:

 {{{-7sqrt(2*(11*11))}}}

The pair of like factors {{{11*11}}} can be written as 
a square {{{11^2}}}

 {{{-7sqrt(2*11^2)}}}

Take individual square roots:

{{{-7sqrt(2)*sqrt(11^2)}}}

The square root of the square of 11 gives only 11:

{{{-7sqrt(2)*11}}}

{{{-77sqrt(2)}}}

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3 SQRT 75

{{{3 sqrt(75)}}}

Break {{{75}}} down into a product of prime numbers:

{{{75=3*5*5}}}

{{{3sqrt(3*5*5)}}}

Group the pair of like factors:

 {{{3sqrt(3*(5*5))}}}

The pair of like factors {{{5*5}}} can be written as 
a square {{{5^2}}}

 {{{3sqrt(3*(5^2))}}}

Take individual square roots:

{{{3sqrt(3)sqrt(5^2)}}}

The square root of the square of 5 gives only 5:

{{{3sqrt(3)*5}}}

{{{15sqrt(3)}}}

Edwin</pre>