Rationalize each expression by building perfect nth root factors for each denominator. Assume all variables represent positive quantities. I don’t understand how to compute these. Also I don’t have the square root sign so I typed it out where it would be and the number before or after.

1. -4/Square root 20
2. Square root of 125/12n^3

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

4. 7+ square root 6 /3-3 square root 2

-4/√20 = -4√20/20 = -1/5 √20 = -2/5 √5

√(125/n^3) = √(25/n^2) √(5/n) = 5/n √(5/n)
= 5/n √(5n/n^2) = 5/n^2 √(5n)

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

(7+√6)/(3-√2)
= (7+√6)(3+√2) / (9-2)
= (21+3√6+7√2+√12)/7
= 1/7 (21 + 3√6 + 7√2 + 2√3)

8*/81-6/16

please help

1/3 + 3/x 1/9 - 9/x square

To rationalize an expression, we want to get rid of any square root signs in the denominator by multiplying both the numerator and denominator by a suitable expression that will eliminate the square root.

Let's go through each expression step by step:

1. -4/Square root 20
To rationalize the denominator, we need to find a perfect square that can be factored out of 20. We see that 4 is a perfect square, so we can express the square root of 20 as the square root of 4 times the square root of 5. The expression becomes:
-4/ (square root 4 * square root 5)

Since the square root of 4 equals 2, we can further simplify this expression to:
-4/(2 * square root 5)
Simplifying even further, we have:
-2/square root 5

2. Square root of 125/12n^3
Similar to the previous example, let's focus on the denominator. We do not have any square root signs in the denominator here, but we can still simplify by breaking down 12 into its prime factorization, which is 2 * 2 * 3. The expression becomes:
(square root of 125)/ (square root of (2^2 * 3) * n^3)

We can then simplify the expression as follows:
(square root of 125)/(2 * square root of (3) * n^3)

Remember that the square root of 125 is 5, so we have:
5/(2 * square root of (3) * n^3)

3. (Square root 10-3) / (Square root 3 + Square root 2)
In this expression, we have square roots in both the numerator and denominator. To rationalize, we need to multiply both the numerator and denominator by the conjugate of the denominator. The conjugate of a binomial is obtained by changing the sign between the terms. So, the conjugate of "Square root 3 + Square root 2" is "Square root 3 - Square root 2". The expression becomes:
[(Square root 10-3) * (Square root 3 - Square root 2)] / [(Square root 3 + Square root 2) * (Square root 3 - Square root 2)]

Expanding the numerator and denominator, we have:
[(Square root 10)(Square root 3) - (Square root 10)(Square root 2) - 3(Square root 3) + 3(Square root 2)] / [(Square root 3)(Square root 3) - (Square root 3)(Square root 2) + (Square root 3)(Square root 2) - (Square root 2)(Square root 2)]

Simplifying further:
[(Square root 30 - Square root 20 - 3(Square root 3) + 3(Square root 2)] / [3 - (Square root 3)(Square root 2) + (Square root 3)(Square root 2) - 2]

Cancelling out some terms, we get:
(Square root 30 - Square root 20 - 3(Square root 3) + 3(Square root 2)] / [1 - 2]
(Square root 30 - Square root 20 - 3(Square root 3) + 3(Square root 2)] / [-1]

Therefore, the simplified rationalized expression is:
-(Square root 30 - Square root 20 - 3(Square root 3) + 3(Square root 2)]

4. 7 + square root 6 / (3 - 3 square root 2)
Similar to the previous example, we have square roots in the denominator, and we need to rationalize them by multiplying both the numerator and denominator by the conjugate of the denominator, which is (3 + 3 square root 2).

The expression becomes:
(7 + square root 6) * (3 + 3 square root 2) / [(3 - 3 square root 2)(3 + 3 square root 2)]

Expanding the numerator and denominator, we have:
[(7)(3) + (7)(3 square root 2) + (square root 6)(3) + (square root 6)(3 square root 2)] / [(3)(3) - (3 square root 2)(3) + (3 square root 2)(3) - (3 square root 2)(3 square root 2)]

Simplifying further:
[21 + 21 square root 2 + 3 (square root 6) + 3 square root 12] / [9 - 9 square root 2 + 9 square root 2 - 18]

Cancelling out some terms, we get:
[21 + 21 square root 2 + 3 (square root 6) + 3 (square root 12)] / [-9]

Therefore, the simplified rationalized expression is:
-(21 + 21 square root 2 + 3 square root 6 + 3 square root 12) / 9

Note: The final simplified expressions may still have some room for further simplification, but these are the rationalized forms of the original expressions.