How do you multiply and simplify #3\sqrt { 50f ^ { 2} g ^ { 3} } \cdot \sqrt { 63f g }#?

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Answer 1 · 2017-05-27T21:59:10

See a solution process below:

Use this rule for multiplying radicals to combine the radicals:

#sqrt(color(red)(a)) * sqrt(color(blue)(b)) = sqrt(color(red)(a) * color(blue)(b))#

#3sqrt(color(red)(50f^2g^3)) * sqrt(color(blue)(63fg)) => 3sqrt(color(red)(50f^2g^3) * color(blue)(63fg)) =>#

#3sqrt((50 * 63)(f^2 * f)(g^3 * g)) =>#

#3sqrt(3150(f^2 * f)(g^3 * g))#

We can now use these rules of exponents to combine the #g# terms:

#a = a^color(red)(1)# and #x^color(red)(a) xx x^color(blue)(b) = x^(color(red)(a) + color(blue)(b))#

#3sqrt(3150(f^2 * f)(g^3 * g)) => 3sqrt(3150(f^2 * f)(g^color(red)(3) * g^color(blue)(1))) =>#

#3sqrt(3150(f^2 * f)g^4)#

We can now rewrite this expression again using the rules for radicals from above in reverse:

#3sqrt(3150(f^2 * f)g^4) => 3sqrt((9 * 25 * 14)(f^2 * f)g^4) =>#

#3sqrt(9 * 25 * f^2 * g^4 * 14f) =>#

#3sqrt(9) * sqrt(25) * sqrt(f^2) * sqrt(g^44) * sqrt(14f) =>#

#(3 * 3 * 5 * f * g^2)sqrt(14f) =>#

#45fg^2sqrt(14f)#