Please show work! For some reason, I keep simplifying it down to 2 and I don't know how to make it in the form of a+b√3. Thanks!. Created by Helena. science-mathematics-en - mathematics-en

Express 1-(1/(1+ √3))+(1/(1- √3)) in the form a+b√3, where a and b are rational numbers?

May 2021 4 119 Report

Express 1-(1/(1+ √3))+(1/(1- √3)) in the form a+b√3, where a and b are rational numbers?

Please show work! For some reason, I keep simplifying it down to 2 and I don't know how to make it in the form of a+b√3. Thanks!

1-(1/(1+ √3))+(1/(1- √3))

1 - (1 - √3)/(1 - √3)(1/(1 + √3) + (1 + √3)/(1 + √3)(1/(1 - √3)

1 - (1 - √3)/(1 - 3) + (1 + √3)/(1 - 3)

1 - (1 - √3)/(-2) + (1 + √3)/(-2)

Distribute the -1

1 +(-1+√3)/(-2) + (1 + √3)/(-2)

1+ (2√3)/(-2)

1+ -√3

The first answer's method is correct, though there's currently an arithmetic mistake in it:

1 - (1 - √3)/(1 - 3) + (1 + √3)/(1 - 3) =

1 - (1 - √3 + 1 + √3)/(-2)

should be

1 - (1 - √3)/(1 - 3) + (1 + √3)/(1 - 3) =

1 - (1 - √3 - 1 - √3)/(-2)

= 1 + (2√3)/(-2)

= 1 - √3

1 - (1 - √3)/(1 - √3)(1/(1 + √3) + (1 + √3)/(1 + √3)(1/(1 - √3) =

1 - (1 - √3)/(1 - 3) + (1 + √3)/(1 - 3) =

1 - (1 - √3 + 1 + √3)/(-2) =

1 + (2/2) =

I suppose it's in the form a + b√3 with a = 2 and b = 0.