How do you multiply xsqrt(10x)*7sqrt(15x)?

1 Answer
Mar 3, 2018

The result is 35x^2sqrt6.

Explanation:

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.