multiplying sq rts

sqrt18a^7b times sqrt27a^8b^6

I think I need to collect like terms such as the a's and b's. My big problem is order of operations. Every problem I do looks so different.

18 = 9 * 2 then sqrt 18 = 3sqrt2

sqrt a^7 = a^3 * sqrt a

27 = 9 * 3 then sqrt 27 = 3 sqrt3
sqrt a^8 = a^4
sqrt b^6 = b^3

Now just multiply the like-terms together.

3sqrt6 and sqrt2a^4

... 3sqrt6 and sqrt2a^4?

So you simplified the problem first as I said above and you get:

3(a^3)sqrt(2a) * 3(a^4)(b^3)sqrt(3)

and then you multiply those 2 things together and you get:

9(a^7)(b^3)sqrt(6a)

To multiply square roots, you can follow these steps:

Step 1: Simplify the expression inside each square root.
Step 2: Multiply the simplified expressions together.
Step 3: Simplify the result if possible.

Let's apply these steps to the given expression: sqrt(18a^7b) times sqrt(27a^8b^6).

Step 1: Simplify the expression inside each square root.
The expression inside the first square root, sqrt(18a^7b), can be simplified as follows:
sqrt(18a^7b) = sqrt(9 * 2 * a^6 * a * b) = 3a^3 * sqrt(2ab)
Similarly, the expression inside the second square root, sqrt(27a^8b^6), can be simplified as follows:
sqrt(27a^8b^6) = sqrt(9 * 3 * a^6 * a^2 * b^6) = 3a^4b^3 * sqrt(3a^2)

Step 2: Multiply the simplified expressions.
Multiply the expressions we obtained in Step 1:
(3a^3 * sqrt(2ab)) * (3a^4b^3 * sqrt(3a^2))

To multiply these expressions, multiply the coefficients (numbers) together and multiply the variables together:
3 * 3 * a^3 * a^4 * b^3 * sqrt(2ab) * sqrt(3a^2)

Simplifying further:
9 * a^7 * b^3 * sqrt(6a^3b)

Step 3: Simplify the result.
Finally, the expression 9 * a^7 * b^3 * sqrt(6a^3b) is the simplified result of multiplying the square roots.

Remember to always simplify the expression inside each square root before multiplying. Also, be careful with the order of operations, multiplying the numerical coefficients together, followed by the variables and then the square roots.