simplifying square roots

1) find volume of this prism?
v=bwh

b=√12"
w=√6"
h=√8"

2)simplify: √576

3)simplify: √400

4)simplify: √256

5)simplify: √(1/4)

ex 2√5 + 6√5= 8√5

the answer has to be in √

but i don't understand it i got help by my teacher but i'm still confused

1)

V = b w h

V = sqrt ( 12 ) * sqrt ( 6 ) * sqrt ( 8 ) = sqrt ( 12 * 6 * 8 ) = sqrt ( 576 ) = 24 " ^ 3

2 )

576 = 24 * 24 = 24 ^ 2

sqrt ( 576 ) = sqrt ( 24 ^ 2 ) = + OR - 24

3 )

400 = 20 * 20 = 20 ^ 2

sqrt ( 400 ) = sqrt ( 20 ^ 2 ) = + OR - 20

4 )

256 = 16 * 16 = 16 ^ 2

sqrt ( 256 ) = sqrt ( 16 ^ 2 ) = + OR - 16

5 )

1 / 4 = ( 1 / 2 ) * ( 1 / 2 ) = ( 1 / 2 ) ^ 2

sqrt ( 1 / 4 ) = sqrt [ ( 1 / 2 ) ^ 2 ] = + OR - 1 / 2

for #1 why did you get a ^ isn't the final answer supposed to be a sqrt

That is maximum simplified answers.

To simplify square roots, you need to find the largest perfect square that is a factor of the number inside the square root symbol. Here's how you can simplify the given square roots:

1) To find the volume of the prism using the formula v = bwh, you need to substitute the given values:
- b = √12" = √4*√3" = 2√3"
- w = √6"
- h = √8" = √4*√2 = 2√2"

Therefore, the volume of the prism is v = 2√3" * √6" * 2√2" = 4√3" * √6" * √2" = 8√(3*6*2)" = 8√(36)" = 8(6)" = 48".

2) To simplify √576, you need to find the largest perfect square that is a factor of 576. In this case, the largest perfect square is 24 since 24 * 24 = 576. Therefore, √576 = √(24*24) = 24.

3) To simplify √400, you need to find the largest perfect square that is a factor of 400. In this case, the largest perfect square is 20 since 20 * 20 = 400. Therefore, √400 = √(20*20) = 20.

4) To simplify √256, you need to find the largest perfect square that is a factor of 256. In this case, the largest perfect square is 16 since 16 * 16 = 256. Therefore, √256 = √(16*16) = 16.

5) To simplify √(1/4), you need to simplify the fraction first. Since 1/4 is already simplified, you can simply take the square root of each number separately. √1 = 1 and √4 = 2. Therefore, √(1/4) = 1/2.

For the example 2√5 + 6√5 = 8√5, it involves combining like terms. Since both terms have the same radical part (√5), you can add their coefficients, which are 2 and 6, to get 8. Therefore, the simplified form is 8√5.

I hope this helps clarify the process of simplifying square roots for you. Let me know if you have any further questions!