Radical and factor expression?

Question

how to rationalize the denominator?

Please show me step by step.

2sqrt(3)/sqrt(5)

How to factor that expression?

1. 8x^3-27y^3

2. 64x^3+125y^3

3. X^4+216x

4. 15x^2-2x-8

5. x^4-13x^2+36

Please show me how to factor those problem

step by step.

Thank You!

Answer 1

2sqrt(3)/sqrt(5). Multiply top and bottom by √5 to get 2√3*√5/5

You could write this as (2/5)√3*√5

Remember that

a3 + b3 = (a + b)(a2 – ab + b2)

a3 – b3 = (a – b)(a2 + ab + b2)

and apply these formulas to 1, 2 and 3. Note that X^4+216x = x(x³+6³)

1. 8x^3-27y^3

2. 64x^3+125y^3

3. X^4+216x

4. 15x^2-2x-8 = (5x-4)(3x+2)

5. x^4-13x^2+36 = (x² - 9)(x² - 4) = (x-3)(x+3)(x-2)(x+2)