Could you please check my answers and help with the ones I can't figure out-

1. Solution to the radical function sqrt(12x+1) = 1 is x = 0
True or False I think it is true according to my calculations

2.Solve x = sqrt(8x)
8x = x^2
x^2 - 8x = 0
x(x-8) = 0
x = 0
x-8 =0
x=8
the only solution is x = 0, correct or no?
3. Simply (3x^2y^-2)^-2/9x^-5y
a.xy^3
b. xy^3/3
xy^3/9
d. xy^3/81

I think it is xy^3/81 but I'm not sure of my calculations
4. Solve 4sqrt(6x-2)>4
a. 1/3<=x<43
b. 1/3<x<43
x>=43
x>43
I think x>43
Thank you

1. To solve the equation sqrt(12x+1) = 1:

Step 1: Isolate the radical term by subtracting 1 from both sides of the equation:
sqrt(12x+1) - 1 = 0

Step 2: Square both sides of the equation to eliminate the square root:
(sqrt(12x+1) - 1)^2 = 0^2
12x + 1 - 2(sqrt(12x+1)) + 1 = 0
12x - 2(sqrt(12x+1)) = 0

Step 3: Divide both sides of the equation by 2 to simplify:
6x - sqrt(12x + 1) = 0

Step 4: Move the square root term to the right side:
sqrt(12x + 1) = 6x

Step 5: Square both sides of the equation again to eliminate the square root:
( sqrt(12x + 1) )^2 = (6x)^2
12x + 1 = 36x^2
36x^2 - 12x - 1 = 0

Now you can solve this quadratic equation. To check if x = 0 is a valid solution, substitute it back into the original equation and see if it satisfies it. If it does, then the statement "x = 0" is true.

2. To solve x = sqrt(8x):

Step 1: Square both sides of the equation to eliminate the square root:
(x)^2 = ( sqrt(8x) )^2
x^2 = 8x

Step 2: Move all terms to one side of the equation to set it equal to zero:
x^2 - 8x = 0

Step 3: Factor out the common factor x:
x(x - 8) = 0

Now set each factor equal to zero and solve for x:
x = 0 or x - 8 = 0

Solving the second equation:
x - 8 = 0
x = 8

Therefore, the solutions are x = 0 and x = 8.

3. To simplify (3x^2y^-2)^(-2)/(9x^-5y):

Step 1: Simplify the exponents:
(3x^2y^(-2))^(-2) = 3^(-2) x^(2*-2) y^(-2*-2) = 1/(3^2 x^(-4) y^(4))
9x^(-5)y is already in a simplified form.

Step 2: Combine the terms:
(1/(3^2 x^(-4) y^(4))) / (9x^(-5)y)
= (1/(9 * 3^2 * x^(-4) * y^(4))) / (9x^(-5)y)
= 1 / (9 * 3^2 * x^(-4) * y^(4) * 9x^(-5)y)
= 1 / (729 * x^(-4) * y^(4) * x^(-5) * y)
= 1 / (729 * x^(-4-5) * y^(4+1))
= 1 / (729 * x^(-9) * y^5)

So the simplified expression is xy^3 / 81 (Option d).

4. To solve 4sqrt(6x-2) > 4:

Step 1: Divide both sides of the inequality by 4 to isolate the square root:
sqrt(6x - 2) > 1

Step 2: Square both sides of the inequality to eliminate the square root. Remember that when squaring an inequality, you need to consider the possibility of introducing extraneous solutions:
( sqrt(6x - 2) )^2 > (1)^2
6x - 2 > 1
6x > 1 + 2
6x > 3

Step 3: Divide both sides of the inequality by 6 to solve for x:
x > 3/6
x > 1/2

Therefore, the solution is x > 1/2 (Option d).