Rationalising the Denominator - Surds (1 Viewer)

[k8s]

Suppose that a, b, c and d are positive integers and c is not a square.
Given that a/(b + sqrt(c)) + d/sqrt(c) is rational, prove that b^2 * d = c (a + d).

--> After getting the common denominator, I tried rationalising the denominator of a/(b + sqrt(c)) + d/sqrt(c) to remove sqrt(c) but I got stuck. Thanks for your help.

 

[ExtremelyBoredUser]

Suppose that a, b, c and d are positive integers and c is not a square.
Given that a/(b + sqrt(c)) + d/sqrt(c) is rational, prove that b^2 * d = c (a + d).

--> After getting the common denominator, I tried rationalising the denominator of a/(b + sqrt(c)) + d/sqrt(c) to remove sqrt(c) but I got stuck. Thanks for your help.

is rational, prove that

tried latexing it but seems like latex broke.

 

[ExtremelyBoredUser]

Suppose that a, b, c and d are positive integers and c is not a square.
Given that a/(b + sqrt(c)) + d/sqrt(c) is rational, prove that b^2 * d = c (a + d).

--> After getting the common denominator, I tried rationalising the denominator of a/(b + sqrt(c)) + d/sqrt(c) to remove sqrt(c) but I got stuck. Thanks for your help.

 

[k8s]

Legend. Thanks for that. Makes perfect sense now.

 

[5uckerberg]

Here is my working out but there are a few things to note

A few common missteps are
1. Not simplifying when given the chance.
2. Stating that because 0 is a rational number and that is irrational.

Note I used 1 on the other side instead of because if things are equal to 1 then it is clear when we perform the algebraic transitions.