How do you rationalize the denominator and simplify $1/(sqrt2-sqrt8)$ ?

Question

1356 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2018-03-06T23:51:49

See a solution process below:

Multiply the fractions by this form of $1$ to rationalize the denominator by removing the radicals from the denominator:

$(color(red)(sqrt(2)) + color(red)(sqrt(8)))/(color(red)(sqrt(2)) + color(red)(sqrt(8)))$

This gives:

$(color(red)(sqrt(2)) + color(red)(sqrt(8)))/(color(red)(sqrt(2)) + color(red)(sqrt(8))) xx 1/(sqrt(2) - sqrt(8)) =>$

$(sqrt(2) + sqrt(8))/((color(red)(sqrt(2)) + color(red)(sqrt(8))) xx (sqrt(2) - sqrt(8))) =>$

$(sqrt(2) + sqrt(8))/((color(red)(sqrt(2))sqrt(2) - color(red)(sqrt(2))sqrt(8) + color(red)(sqrt(8))sqrt(2) - color(red)(sqrt(8))sqrt(8))) =>$

$(sqrt(2) + sqrt(8))/(sqrt(2)^2 - color(red)(sqrt(2))sqrt(8) + sqrt(2)color(red)(sqrt(8)) - sqrt(8)^2) =>$

$(sqrt(2) + sqrt(8))/(2 - 0 - 8) =>$

$(sqrt(2) + sqrt(8))/(-6) =>$

$-(sqrt(2) + sqrt(8))/6$

How do you rationalize the denominator and simplify 1/(sqrt2-sqrt8)1/(sqrt2-sqrt8)?