1. √¯8y^5 * √¯40y^2
2. 4√¯2x * 5√¯6xy^2
3
3. √¯2x^2y^2 * 2√¯15x^5y
2. 4√¯2x * 5√¯6xy^2
3
3. √¯2x^2y^2 * 2√¯15x^5y

Multiply and simplify. assume that all variables are positive?
1. √¯8y^5 * √¯40y^2
2y²√2y * 2y√10
4y³√20y
8y³√5y
2. 4√¯2x * 5√¯6xy^2
4√2x * 5y√6x
20y√12x²
40xy√3
3. √¯2x^2y^2 * 2√¯15x^5y
xy√2 * 2x²√15xy
2x³y√30xy
1. √¯8y^5 * √¯40y^2
2y²√2y * 2y√10
4y³√20y
8y³√5y
2. 4√¯2x * 5√¯6xy^2
4√2x * 5y√6x
20y√12x²
40xy√3
3. √¯2x^2y^2 * 2√¯15x^5y
xy√2 * 2x²√15xy
2x³y√30xy

Ok, the rules with roots is that you can multiply anything with the same root. Right now, you are only working with ^2√ roots and you have no combinations, such as ^2√x * ^3√y. Ok, now that we're past the principle of the matter, we can move onto the math.
Your original equation
√¯8y^5 * √¯40y^2
Actually, What I like to do (if I can) is simplify first and then multiply so i'm not working with some variable to the seventh power (^7) or some obscurely large number. Here's what I did to simplify:
(2*y^2) * (√¯2y) * (2y) * (√¯10)
Do you see how I did that? all i did was i took everything that could be square rooted and I moved it outside of the root sign (√). Next, What i'm going to do is rewrite everything with exponents and not with root signs:
(2*y^2) * [(2y)^(1/2)] * (2y) * [(10)^(1/2)]
Next, I'm going to group everything that has like terms ie. Things that do not have a half root inside of the parentheses and things that do with each other respectively:
(4*y^3) * [(20y)^(1/2)]
Now, you can simplify one more time by taking a four out from the root ie (...^(1/2)) Like so:
(8*y^3) * [(5y)^(1/2)] or (8*y^3) * √(5y)
This is the term that you should be left with by the end of your calculation. Everything in this answer is prime if it is the root sign and simplified if it is outside of the root sign.
Now, the second calculation can be done the exact same way. For this one, I am assuming that the numbers in front of the root signs are coefficients and not the power that the root sign is to otherwise that could jeopardize this entire endeavor that I am about to embark on. I will not be explaining my every step on the following, however I follow a couple of general rules:
1) try to simplify by taking everything out of the roots that you can
2) Group like terms
#2
4√¯2x * 5√¯6xy^2
(20) * √2x * (y) * √6x
20y * √12x^2
20y * 2x * √3
40xy * √3
#3
√¯2x^2y^2 * 2√¯15x^5y
2 * x^3 * y * √2 * √15xy
2yx^3 * √(30xy)
Your original equation
√¯8y^5 * √¯40y^2
Actually, What I like to do (if I can) is simplify first and then multiply so i'm not working with some variable to the seventh power (^7) or some obscurely large number. Here's what I did to simplify:
(2*y^2) * (√¯2y) * (2y) * (√¯10)
Do you see how I did that? all i did was i took everything that could be square rooted and I moved it outside of the root sign (√). Next, What i'm going to do is rewrite everything with exponents and not with root signs:
(2*y^2) * [(2y)^(1/2)] * (2y) * [(10)^(1/2)]
Next, I'm going to group everything that has like terms ie. Things that do not have a half root inside of the parentheses and things that do with each other respectively:
(4*y^3) * [(20y)^(1/2)]
Now, you can simplify one more time by taking a four out from the root ie (...^(1/2)) Like so:
(8*y^3) * [(5y)^(1/2)] or (8*y^3) * √(5y)
This is the term that you should be left with by the end of your calculation. Everything in this answer is prime if it is the root sign and simplified if it is outside of the root sign.
Now, the second calculation can be done the exact same way. For this one, I am assuming that the numbers in front of the root signs are coefficients and not the power that the root sign is to otherwise that could jeopardize this entire endeavor that I am about to embark on. I will not be explaining my every step on the following, however I follow a couple of general rules:
1) try to simplify by taking everything out of the roots that you can
2) Group like terms
#2
4√¯2x * 5√¯6xy^2
(20) * √2x * (y) * √6x
20y * √12x^2
20y * 2x * √3
40xy * √3
#3
√¯2x^2y^2 * 2√¯15x^5y
2 * x^3 * y * √2 * √15xy
2yx^3 * √(30xy)

no idea sorry
love ya any way
love ya any way