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.

Thanks!

I'd be happy to help you with these questions and explain the steps to get to the final answers.

1. To rationalize the expression 3/square root 12, we can multiply the numerator and denominator by the conjugate of the denominator, which is square root 12.

(3/square root 12) * (square root 12/square root 12)

= (3 * square root 12) / 12

= (3 * square root (4 * 3)) / 12

= (3 * square root 4 * square root 3) / 12

= (3 * 2 * square root 3) / 12

= (6 * square root 3) / 12

= (square root 3) / 2

2. To rationalize the expression square root 27/50b, we can multiply the numerator and denominator by the conjugate of the numerator, which is square root 27.

(square root 27/50b) * (square root 27/square root 27)

= (square root (27 * 27)) / (50b * square root 27)

= (27/ (50b * square root 27))

= (27 * square root 27) / (50b * square root (3 * 3 * 3))

= (27 * 3 * square root 3) / (50b * 3)

= (9 * square root 3) / (50b)

= 9 square root 3 / 50b

3. To rationalize the expression -4/square root 20, we can multiply the numerator and denominator by the conjugate of the denominator, which is square root 20.

(-4/square root 20) * (square root 20/square root 20)

= (-4 * square root 20) / (20)

= (-4 * square root (4 * 5)) / (20)

= (-4 * square root 4 * square root 5) / (20)

= (-4 * 2 * square root 5) / (20)

= (-8 * square root 5) / (20)

= -2 * square root 5 / 5

4. To rationalize the expression square root 125/12n^3, we can multiply the numerator and denominator by the conjugate of the numerator, which is square root 125.

(square root 125/12n^3) * (square root 125/square root 125)

= (square root (125 * 125)) / (12n^3 * square root 125)

= (125/ (12n^3 * square root 125))

= (125 * square root 125) / (12n^3 * square root (5 * 5 * 5))

= (125 * 5 * square root 5) / (12n^3 * 5)

= (25 * square root 5) / (24n^3)

Now let's work on the simplification of expressions:

5. To simplify 8/3 + square root 11, we cannot rationalize the denominator in this case. So, the expression remains as is.

6. To simplify 6/square root x - square root 2, we can rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator, which is square root x + square root 2.

(6 / (square root x - square root 2)) * ((square root x + square root 2) / (square root x + square root 2))

= (6 * (square root x + square root 2)) / (x - 2)

7. To simplify 7/ square root 7 + 3, we can rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator, which is square root 7 - 3.

(7 / (square root 7 + 3)) * ((square root 7 - 3) / (square root 7 - 3))

= (7 * (square root 7 - 3)) / (-2)

8. To simplify square root 10 - 3 / square root 3 + square root 2, we can rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator, which is square root 3 - square root 2.

((square root 10 - 3) / (square root 3 + square root 2)) * ((square root 3 - square root 2) / (square root 3 - square root 2))

= ((square root 10 - 3) * (square root 3 - square root 2)) / (1)

= (square root 30 - square root 20 - 3 * square root 3 + 3 * square root 2)

9. To simplify 7 + square root 6/ 3-3 square root 2, we cannot rationalize the denominator in this case. So, the expression remains as is.

I hope this helps you understand the steps required to solve these problems and rationalize expressions. Let me know if you have any further questions!

Sure! I'll provide you with the step-by-step solutions for each question and explain how we got to each step.

1. Rationalize the expression: 3/Square root 12

Step 1: To rationalize the denominator, we need to multiply both the numerator and denominator by the conjugate of the denominator. The conjugate of the square root of 12 is (- square root 12).

Step 2: Multiply the numerator and denominator by (- square root 12):

3 * (- square root 12) / (square root 12 * (- square root 12))

Step 3: Simplify the expression:

-3 square root 12 / 12

Step 4: Further simplify the expression:

- square root 12 / 4

Final answer: - square root 3 / 2

2. Rationalize the expression: Square root 27/50 b

Step 1: To rationalize the denominator, we need to multiply both the numerator and denominator by the conjugate of the denominator. The conjugate of 50b is 50b.

Step 2: Multiply the numerator and denominator by 50b:

(square root 27 * 50b) / (50b * 50b)

Step 3: Simplify the expression:

50b square root 27 / (2500b^2)

Final answer: 50b square root 27 / (2500b^2)

3. Rationalize the expression: -4/square root 20

Step 1: To rationalize the denominator, we need to multiply both the numerator and denominator by the conjugate of the denominator. The conjugate of the square root of 20 is (- square root 20).

Step 2: Multiply the numerator and denominator by (- square root 20):

-4 * (- square root 20) / (square root 20 * (- square root 20))

Step 3: Simplify the expression:

4 square root 20 / 20

Step 4: Further simplify the expression:

2 square root 20 / 10

Final answer: square root 20 / 5

4. Rationalize the expression: square root 125/12n^3

Step 1: To rationalize the denominator, we need to multiply both the numerator and denominator by the conjugate of the denominator. The conjugate of 12n^3 is 12n^3.

Step 2: Multiply the numerator and denominator by 12n^3:

(square root 125 * 12n^3) / (12n^3 * 12n^3)

Step 3: Simplify the expression:

12n^3 square root 125 / (144n^6)

Step 4: Further simplify the expression:

12n^3 * 5 square root 5 / (144n^6)

Final answer: 60n^3 square root 5 / (144n^6)

5. Simplify the expression: 8/3 + square root 11

There is no rationalizing of the denominator required in this question.

No further simplification is possible.

Final answer: 8/3 + square root 11

6. Simplify the expression: 6/square root x - square root 2

Step 1: Rationalize the denominator. Multiply both the numerator and denominator by the conjugate of the denominator. The conjugate of the square root of x is (square root x).

Step 2: Multiply the numerator and denominator by (square root x):

(6 * square root x) / (square root x * square root x) - (square root 2 * square root x)

Step 3: Simplify the expression:

6 square root x / x - square root 2x

No further simplification is possible.

Final answer: 6 square root x / (x - square root 2x)

7. Simplify the expression: 7/ square root 7 + 3

Step 1: Rationalize the denominator. Multiply both the numerator and denominator by the conjugate of the denominator. The conjugate of the square root of 7 is (square root 7).

Step 2: Multiply the numerator and denominator by (square root 7):

(7 * square root 7) / (square root 7 * square root 7) + (3 * square root 7)

Step 3: Simplify the expression:

7 square root 7 / 7 + 3 square root 7

Step 4: Combine like terms:

square root 7 (7 + 3)

Step 5: Further simplify the expression:

10 square root 7

No further simplification is possible.

Final answer: 10 square root 7

8. Simplify the expression: Square root 10 - 3 / square root 3 + square root 2

Step 1: Rationalize the denominator. Multiply both the numerator and denominator by the conjugate of the denominator. The conjugate of (square root 3 + square root 2) is (square root 3 - square root 2).

Step 2: Multiply the numerator and denominator by (square root 3 - square root 2):

(square root 10 - 3) * (square root 3 - square root 2) / ((square root 3 + square root 2) * (square root 3 - square root 2))

Step 3: Simplify the expression:

(square root 10 * square root 3) - (3 * square root 3) - (square root 10 * square root 2) + 3 * square root 2 / (3 - 2)

Step 4: Simplify further:

square root 30 - 3 square root 3 - square root 20 + 3 square root 2

No further simplification is possible.

Final answer: square root 30 - 3 square root 3 - square root 20 + 3 square root 2

9. Simplify the expression: 7 + square root 6 / 3 - 3 square root 2

Step 1: Rationalize the denominator. Multiply both the numerator and denominator by the conjugate of the denominator. The conjugate of (3 - 3 square root 2) is (3 + 3 square root 2).

Step 2: Multiply the numerator and denominator by (3 + 3 square root 2):

(7 + square root 6) * (3 + 3 square root 2) / ((3 - 3 square root 2) * (3 + 3 square root 2))

Step 3: Simplify the expression:

(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 square root 2))

Step 4: Simplify further:

21 + 21 square root 2 + 3 square root 6 + 3 square root 12 / (9 - 18)

Step 5: Simplify further:

21 + 21 square root 2 + 3 square root 6 + 3 square root 12 / (- 9)

No further simplification is possible.

Final answer: (21 + 21 square root 2 + 3 square root 6 + 3 square root 12) / (- 9)

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