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Simplifying Rational and Radical Expressions

1. Simplify the rational expression:
2x^2 - 12x + 18/2x^2 - 18

2. Divide:
x^2 - 9x + 20/3x - 15 ÷ x^2 - 16/9x + 36

3. Simplify:
√[81x^12y^8z^10]

4. Perform the indicated operations:
2√[48] + 7√[12] -√[27]

5. Multiply:
(√[6] + 2√[2])(4√[6] - 3√[2])

6. Rationalize the denominator:
2____
√[3] + √[2]

7. Factor by grouping: 3x^2 - 12x + 5x - 20

8. Factor completely: 9x^2 - 16y^2

9. Factor completely: 3x^2 - 2x - 8

10. Factor completely: 3x^2 + 12x +12

11. Simplify the rational expression: 2^2 - 3x - 9/2x^2 + 3x - 9

12. Show the similarities between dividing two fractions and dividing two rational expressions using examples of each; explain your steps.

Solution Preview

1. Simplify the rational expression:
2x^2 - 12x + 18/2x^2 - 18
I think you mean:
(2x^2 - 12x + 18) / (2x^2 - 18)
= 2(x^2-6x + 9)/ 2(x^2 - 9)
= (x-3)^2 / (x+3)(x-3)
= (x-3) / (x+3)

2. Divide:
x^2 - 9x + 20/3x - 15 ÷ x^2 - 16/9x + 36
= (x^2 - 9x + 20)/(3x - 15) ÷ (x^2 - 16)/(9x + 36)
= (x-4)(x-5)/3(x-5) ÷ (x+4)(x-4)/9(x+4)
= (x-4)/3 ÷(x-4)/9
= 9/3
= 3

3. Simplify:
√[81x^12y^8z^10]
= √[9^2 (x^6)^2 (y^4)^2 (z^5)^2]
= [9^2 (x^6)^2 (y^4)^2 (z^5)^2]^0.5
= 9(x^6)(y^4)(z^5)

4. Perform the indicated ...

Solution Summary

Rational and radical expressions are simplified. The solution is detailed and well presented. The response was given a rating of "5/5" by the student who originally posted the question.

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