Question

How do you simplify `(3+ isqrt2)/ (7-isqrt2)`?

Answer 1

Multiplying denominator and numerator by the complex conjugate of the numerator, `(7+isqrt2)`, shows that this expression can be simplified to: `(21+10isqrt2-2)/(51)`

Explanation:

The complex conjugate of a number `(a+bi)` is `(a-bi)`, and multiplying top (denominator) and bottom (numerator) of a complex fraction by the complex conjugate of the bottom (numerator) will simpifly it:

`(3+isqrt2)/(7-isqrt2)*(7+isqrt2)/(7+isqrt2)`

Note that any number, including a complex number, divided by itself is 1, so in multiplying by `(7+isqrt2)/(7+isqrt2)` we are in effect multiplying by 1, leaving the result unchanged.

Note: `i*i=-1` and `sqrt2*sqrt2=2`

`(3+isqrt2)/(7-isqrt2)*(7+isqrt2)/(7+isqrt2)` = `(21+3isqrt2+7isqrt2-2)/(49+7isqrt2-7isqrt2+2)`
= `(21+10isqrt2-2)/(49+2) = (21+10isqrt2-2)/(51)`