Simplifying (4ab x (sqrt 2b)) - (3a x (sqrt18b^3)) + (7ab x (sqrt6b))

I see it as

4ab√(2b) - 3a√(18b^3) + 7ab√(6b)

= 4ab√(2b) - 9ab√(2b) + 7ab√(6b)
= -5ab√(2b) + 7ab√(6b)

we could write √(6b) as √3√(2b) , but really see no advantage , and it certainly wouldn't look more "simplified"

To simplify the expression (4ab x √2b) - (3a x √18b^3) + (7ab x √6b), we need to simplify each individual term and then combine like terms.

First, let's simplify each term:

Term 1: 4ab x √2b
To simplify this, we multiply the coefficients (4 and 2) and the variables (a and b), and then take the square root of b:
= 4 x 2 x a x √(b x b) x √2
= 8ab√2

Term 2: 3a x √18b^3
To simplify this, we multiply the coefficients (3 and 18), the variable a, and the variable b. We also simplify the square root of 18 and b^3:
= 3 x 18 x a x √(b x b x b^2)
= 54ab√(b^2)
= 54ab√b

Term 3: 7ab x √6b
To simplify this, we multiply the coefficients (7 and 6) and the variables (a and b), and then take the square root of b:
= 7 x 6 x a x √(b x b) x √6
= 42ab√6

Now, we can combine the simplified terms:

New Expression: 8ab√2 - 54ab√b + 42ab√6

Note that we cannot further simplify the expression because the terms have different square roots.