how to simplifing radicals?

Question

I am unsure of how to radical signs on the computer.

2 radical 3-4 radical 2+6 radical 3+8 radical 2

2 radical 3 + 3 radical 5 - 4 radical 5

3 radical 12 + 4 radical 18

-2 radical 8 + 2 radical 2 - 3 radical 18

radical 9x^4 + radical 16

radical a^4 + radical a^5

2 radical 12x^3y^3 - radical 3xy

radical 32a^8b +radical 50 a^16b

Answer 1

Add together what has similar square roots.
If you can reduce first, do so, then add them together.
I broke them down so you could see how to do that, then pull it out from under the radical sign.

2√3 - 4√2 + 6√3 + 8√2
= 2√3 + 6√3 - 4√2 + 8√2
= 8√3 + 4√2

2√3 + 3√5 - 4√5
= 2√3 - 1√5
= 2√3 - √5

3√12 + 4√18
= 3√(4*3) + 4√(9*2)
= 3*2√3 + 4*3√2
= 6√3 + 12√2

-2√8 + 2√2 - 3√18
= -2√(4*2) + 2v2 - 3√(9*2)
= -2*2√2 + 2√2 - 3*3√2
= -4√2 + 2√2 - 9√2
= -11√2

√9x^4 + √16
= 3x² + 4

√a^4 + √a^5
=a² + a²√a

2√12x^3y^3 - √3xy
= 2√(4*3x³y³) - √3xy
= 2*2xy√3xy - √3xy
= 4xy√3xy - √3xy
= (4xy - 1)√3xy

√32a^8b +√50 a^16b
= √(16*2*a^8b) + √(25*2a^16b)
= 4a^4√2b + 5a^8√2b
= (4a^4 + 5a^8)√2b

Answer 2

Lets assume radicak = square root =sqrt

So look at 2sqrt(3) - 4sqrt(2)+6sqrt(3)+8sqrt(2)

You can add terms with like sqrts, i.e. 2sqrt(3) + 6sqrt(3) = 8 sqrt(3), so the first equation reduces to:

8sqrt(3)+4sqrt(2)

You can apply that to the next 3.

Fifth problem has at least one other step - sqrt(A*x^n) has to be reduced

If n is even, the sqrt(x^n) = x^n/2. If n is odd, sqrt(x^n) = x^(n-1)/2*sqrt(x)

SO fifth problem is

sqrt(9x^4) + sqrt(16) = 3sqrt(x^4)+4 (9 and 16 are perfect squares)

3sqrt(x^4)+4 =3x^2+4

Apply to remaining problems.