Please show work

Simplify rational

1. 4√(a^6 b^13) / 4√(a^2b)

2.√(2x^3 / 49y^4)

Add or Subtract
3. x√(75xy) - √(27x^3y)

Multiply
4.(√(5) - 5) (2√(5) + 2)

Rationalize the denominator
5. √(25x^5 / 3y^2)

6. 5 / 2-√(7)

do you have any idea how to do any of these or where to start??

I know the basics but I'm still not solid on the concepts. I mainly want to see the processes. Any help would be really appreciated.

Also in number 5 it's 24 not 25 sorry.

It seems that you need a lesson in everything you need to know about radicals.

#3 You can't combine radicals unless they are the same like 3x - 2x = 1x
So yo have to write 75 xy as 25(3xy) you can take the square root of 25 with the 3xy left under the radical. for 27x^3y write it as 9x^2(3xy) Take the square root of 9x^2 and the 3xy is left under the radical. Now. you have like terms. Keep the radical 3xy and combine what is out in front.

#4 multiply that one like you woud (x -5)(2x +2) It works like multiplying binomials. It gets a little more confusing because of the radical. Remember square root of 5 times square root of 5 is 5.
#5 put the radical over the numerator and over the denominator. Multipy the numerator and denominator by the square root of 3y^2. When you multiply square root of 3y^2 times square root of 3y^2 you will get just 3y^2.

#6 multiply the numertor and the denominator by the conjugate which is
2 +(radical 7).

#3 I got 2ab^2 3√(3a^2b)

In #4 I'm not really sure how to multiply the radical is the problem, it's the FOIL method but I still don't really know how to handle it with a radical instead.

Sure! Here are the step-by-step solutions for each problem:

1. Simplify rational expression:
To simplify 4√(a^6 b^13) / 4√(a^2b):
Since the bases are the same, we can divide the exponents of a and b.
√(a^6 / a^2) = √(a^(6-2)) = √(a^4) = a^2
Similarly, √(b^13 / b) = √(b^(13-1)) = √(b^12) = b^6
Therefore, the simplified expression is a^2b^6.

2. Simplify rational expression:
To simplify √(2x^3 / 49y^4):
Split the radical in the numerator and denominator:
√(2x^3) / √(49y^4)
Simplify each radical separately:
√(2)x√(x^3) / 7y^2√(y^4)
√(2)x(x^(3/2)) / 7y^2y^2
Combine like terms and simplify:
(x√(2)x^(3/2)) / (49y^4)
The simplified expression is x√(2)x^(3/2) / 49y^4.

3. Add or Subtract:
To add x√(75xy) - √(27x^3y):
The expression cannot be further simplified since the radicals have different terms.
The final answer is x√(75xy) - √(27x^3y).

4. Multiply:
To multiply (√(5) - 5) (2√(5) + 2):
Apply the distributive property:
√(5) * 2√(5) + √(5) * 2 - 5 * 2√(5) - 5 * 2
Simplify each term:
2√(5)(√(5)) + 2√(5) - 10√(5) - 10
Combine like terms:
2√(5)^2 + 2√(5) - 10√(5) - 10
Simplify the square root:
2 * 5 + 2√(5) - 10√(5) - 10
Multiply and combine like terms:
10 + 2√(5) - 10√(5) - 10
Simplify further:
2 - 8√(5)
The simplified expression is 2 - 8√(5).

5. Rationalize the denominator:
To rationalize the denominator of √(25x^5 / 3y^2):
Multiply the numerator and denominator by the conjugate of the denominator.
√(25x^5 / 3y^2) * (√(3y^2) / √(3y^2))
Simplify each radical separately:
√(25x^5) * √(3y^2) / √(3y^2) * √(3y^2)
Simplify further:
√(75x^5y^2) / (√(3y^2))^2
Simplify the square root:
√(75x^5y^2) / 3y^2
Combine like terms, if possible:
(√(25x^4y^2) * √(3x)) / 3y^2
Simplify the square root:
(5x^2y * √(3x)) / 3y^2
The rationalized expression is (5x^2y * √(3x)) / 3y^2.

6. Rationalize the denominator:
To rationalize the denominator of 5 / (2-√(7)):
Multiply the numerator and denominator by the conjugate of the denominator.
5 / (2-√(7)) * (2+√(7)) / (2+√(7))
Apply the difference of squares formula:
(5(2+√(7))) / (2^2 - (√(7))^2)
Simplify further:
(10 + 5√(7)) / (4 - 7)
Combine like terms and simplify:
(10 + 5√(7)) / (-3)
The rationalized expression is (-10 - 5√(7)) / 3.

These are the step-by-step solutions for each problem. If you have any further questions, feel free to ask!