I need some help with a handful of questions. I need to see the steps on how to get to the final answers. I would also appreciate if you can explain how you got to those steps.

Rationalize each expression by building perfect nth root factors for each denominator. Assume all variables represent positive quantities.

1. 3/Square root 12

2. Square root 27/50 b The square root on this problem covers the whole problems.

3. -4/square root 20

4. square root 125/12n^3 The square root on this question covers the whole problem.

Simplify the following expressions by rationalizing the denominators were possible, state results in exact from and approximate form, rounded to hundredths.

5. 8/3 + square root 11

6. 6/square root x - square root 2

7. 7/ square root 7 + 3

8. Square root 10 - 3 / square root 3 + square root 2

9. 7 + square root 6/ 3-3 square root 2

Thanks for your help my professor is not at all helpful and the book that is provided only explains basic examples rather then some like what we have to work through.

I will do #4 and #8 ,you try the rest and let me know where you run into problems

#4

-4/√20
= -4/(2√5) = -2/√5
= -2/√5 * √5/√5
= -2√5/5 , now check with your calculator to see if correct

#8

(√10 - 3)/(√3 + √2)
= (√10 - 3)/(√3 + √2) * (√3 - √2)/(√3 - √2)
= (√30 - √20 -3√3 - 3√2)/(3-2)
= √30 - 2√5 - 3√3 + 3√2

Which statement is true about the square root of 28?

Sure, I'd be happy to help you step-by-step with your questions. Let's go through each question one by one.

1. Rationalize the expression 3/square root 12:
- We start by simplifying the square root of 12: square root 12 = square root (4 * 3) = square root 4 * square root 3 = 2 * square root 3.
- Now, we can rewrite the expression as 3/(2 * square root 3).
- To rationalize the denominator, we multiply the numerator and denominator by the conjugate of the denominator, which in this case is 2 * square root 3.
- Simplifying, we get (3 * 2 * square root 3) / (2 * square root 3 * 2 * square root 3) = 6 * square root 3 / (2 * 3) = 2 * square root 3.

2. Rationalize the expression square root 27 / (50b):
- We start with the square root of 27: square root 27 = square root (9 * 3) = square root 9 * square root 3 = 3 * square root 3.
- Now, we can rewrite the expression as (3 * square root 3) / (50b).
- To rationalize the denominator, we multiply the numerator and denominator by the conjugate of the denominator, which in this case is 50b.
- Simplifying, we get (3 * square root 3 * 50b) / (50b * 50b) = 150b * square root 3 / (2500b^2) = 3 * square root 3 / (50b).

3. Rationalize the expression -4/square root 20:
- We start by simplifying the square root of 20: square root 20 = square root (4 * 5) = square root 4 * square root 5 = 2 * square root 5.
- Now, we can rewrite the expression as -4/(2 * square root 5).
- To rationalize the denominator, we multiply the numerator and denominator by the conjugate of the denominator, which in this case is 2 * square root 5.
- Simplifying, we get (-4 * 2 * square root 5) / (2 * square root 5 * 2 * square root 5) = -8 * square root 5 / (2 * 5) = -4 * square root 5 / 5.

4. Rationalize the expression square root 125 / (12n^3):
- We start with the square root of 125: square root 125 = square root (25 * 5) = square root 25 * square root 5 = 5 * square root 5.
- Now, we can rewrite the expression as (5 * square root 5) / (12n^3).
- To rationalize the denominator, we multiply the numerator and denominator by the conjugate of the denominator, which in this case is (12n^3).
- Simplifying, we get (5 * square root 5 * 12n^3) / (12n^3 * 12n^3) = 60n^3 * square root 5 / (12n^3 * 12n^3) = 5 * square root 5 / (144n^6).

Now let's move on to simplifying the expressions by rationalizing the denominators wherever possible.

5. Simplify 8/3 + square root 11:
- There are no denominators to rationalize, so there's no need for any additional steps. The expression is already simplified.

6. Simplify 6/square root x - square root 2:
- To rationalize the denominator of the first fraction, we multiply the numerator and denominator by the conjugate of the denominator, which is square root x + square root 2.
- Simplifying, we get (6 * (square root x + square root 2)) / ((square root x * square root x) - (square root 2 * square root x)) = (6 * (square root x + square root 2)) / (x - square root 2x).

7. Simplify 7/square root 7 + 3:
- To rationalize the denominator, we multiply the numerator and denominator by the conjugate of the denominator, which is square root 7 - 3.
- Simplifying, we get (7 * (square root 7 - 3)) / ((square root 7 * square root 7) - (3 * square root 7)) = (7 * (square root 7 - 3)) / (7 - 3 * square root 7).

8. Simplify square root 10 - 3 / square root 3 + square root 2:
- To rationalize the denominator of the second fraction, we multiply the numerator and denominator by the conjugate of the denominator, which is square root 3 - square root 2.
- Simplifying, we get ((square root 10 - 3) * (square root 3 - square root 2)) / ((square root 3 * square root 3) - (square root 2 * square root 3) + (square root 3 * square root 2) - (square root 2 * square root 2)) = ((square root 10 - 3) * (square root 3 - square root 2)) / (3 - square root 6 + square root 6 - 2) = ((square root 10 - 3) * (square root 3 - square root 2)) / (1).

9. Simplify 7 + square root 6 / (3 - 3 square root 2):
- To rationalize the denominator of the second fraction, we multiply the numerator and denominator by the conjugate of the denominator, which is 3 + 3 square root 2.
- Simplifying, we get (7 + square root 6) * (3 + 3 square root 2) / ((3 * 3) - (3 * 3 square root 2) + (3 square root 2) - (3 square root 2 * square root 2)) = (21 + 21 square root 2 + 3 square root 6 + 3 square root 12) / (9 - 6) = (21 + 21 square root 2 + 3 square root 6 + 3 square root 12) / 3 = 7 + 7 square root 2 + square root 6 + square root 12.

I hope this helps! Let me know if you have any further questions.