Use the radical multiplication and simplification rules:
Multiplication: #sqrta*sqrtb=sqrt(ab)#
Simplification: #sqrt(a^2)=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)#
#=color(red)x*color(blue)sqrt(10x)*color(red)7*color(blue)sqrt(15x)#
#=color(red)x*color(red)7*color(blue)sqrt(10x)*color(blue)sqrt(15x)#
#=color(red)(7x)*color(blue)sqrt(10x)*color(blue)sqrt(15x)#
#=color(red)(7x)*color(blue)sqrt(10x*15x)#
#=color(red)(7x)*color(blue)sqrt(10*15*x*x)#
#=color(red)(7x)\*color(blue)sqrt(150*x*x)#
#=color(red)(7x)*color(blue)sqrt(150*x^2)#
Next, use the multiplication rule backwards:
#color(white)=color(red)(7x)*color(blue)sqrt(150*x^2)#
#=color(red)(7x)*color(blue)sqrt(150)*color(blue)(sqrt(x^2))#
Now, use the simplification rule:
#color(white)=color(red)(7x)*color(blue)sqrt(150)*color(blue)(sqrt(x^2))#
#=color(red)(7x)*color(blue)sqrt(150)*color(red)x#
#=color(red)(7x)*color(red)x*color(blue)sqrt(150)#
#=color(red)(7x^2)*color(blue)sqrt(150)#
Technically, this answer is correct, but it can be simplified further by factoring #150# and then using the simplification rule backward again:
#color(white)=color(red)(7x^2)*color(blue)sqrt(150)#
#=color(red)(7x^2)*color(blue)sqrt(6*25)#
#=color(red)(7x^2)*color(blue)sqrt6*color(blue)sqrt25#
#=color(red)(7x^2)*color(blue)sqrt6*color(blue)sqrt(5^2)#
#=color(red)(7x^2)*color(blue)sqrt6*color(red)5#
#=color(red)(7x^2)*color(red)5*color(blue)sqrt6#
#=color(red)(35x^2)*color(blue)sqrt6#
This answer is fully simplified.
