Use the radical multiplication and simplification rules:
Multiplication: sqrta*sqrtb=sqrt(ab)√a⋅√b=√ab
Simplification: sqrt(a^2)=a√a2=a
For this problem, first, multiply the radicals (in blue) and their coefficients (in red) together:
color(white)=color(red)xcolor(blue)sqrt(10x)*color(red)7color(blue)sqrt(15x)=x√10x⋅7√15x
=color(red)x*color(blue)sqrt(10x)*color(red)7*color(blue)sqrt(15x)=x⋅√10x⋅7⋅√15x
=color(red)x*color(red)7*color(blue)sqrt(10x)*color(blue)sqrt(15x)=x⋅7⋅√10x⋅√15x
=color(red)(7x)*color(blue)sqrt(10x)*color(blue)sqrt(15x)=7x⋅√10x⋅√15x
=color(red)(7x)*color(blue)sqrt(10x*15x)=7x⋅√10x⋅15x
=color(red)(7x)*color(blue)sqrt(10*15*x*x)=7x⋅√10⋅15⋅x⋅x
=color(red)(7x)*color(blue)sqrt(150*x*x)=7x⋅√150⋅x⋅x
=color(red)(7x)*color(blue)sqrt(150*x^2)=7x⋅√150⋅x2
Next, use the multiplication rule backwards:
color(white)=color(red)(7x)*color(blue)sqrt(150*x^2)=7x⋅√150⋅x2
=color(red)(7x)*color(blue)sqrt(150)*color(blue)(sqrt(x^2))=7x⋅√150⋅√x2
Now, use the simplification rule:
color(white)=color(red)(7x)*color(blue)sqrt(150)*color(blue)(sqrt(x^2))=7x⋅√150⋅√x2
=color(red)(7x)*color(blue)sqrt(150)*color(red)x=7x⋅√150⋅x
=color(red)(7x)*color(red)x*color(blue)sqrt(150)=7x⋅x⋅√150
=color(red)(7x^2)*color(blue)sqrt(150)=7x2⋅√150
Technically, this answer is correct, but it can be simplified further by factoring 150150 and then using the simplification rule backward again:
color(white)=color(red)(7x^2)*color(blue)sqrt(150)=7x2⋅√150
=color(red)(7x^2)*color(blue)sqrt(6*25)=7x2⋅√6⋅25
=color(red)(7x^2)*color(blue)sqrt6*color(blue)sqrt25=7x2⋅√6⋅√25
=color(red)(7x^2)*color(blue)sqrt6*color(blue)sqrt(5^2)=7x2⋅√6⋅√52
=color(red)(7x^2)*color(blue)sqrt6*color(red)5=7x2⋅√6⋅5
=color(red)(7x^2)*color(red)5*color(blue)sqrt6=7x2⋅5⋅√6
=color(red)(35x^2)*color(blue)sqrt6=35x2⋅√6
This answer is fully simplified.
