Question

idek

Answer 1

Simplify
\[\frac{ 1 + \frac{ 1 }{\sqrt3 } }{ 1 - \frac{ 1 }{ \sqrt3 } }\]

After some trig identities and other fun stuff, I have ended up here. And I can't seem to simplify this correctly. Maybe I've finally burnt my brain with studying for AP exams next week. :shrug: @Hero

Answer 2

@SmokeyBrown

Answer 3

@nuts

Answer 4

difference of squares

Answer 5

Do you have answer choices?

Answer 6

I know the answer, I just don't the proper steps to get there.

Answer 7

Oh

Answer 8

Inky, you feel like showing steps

Answer 9

multiply by numerator/numerator

Answer 10

\[\frac{ (1 + \frac{ \sqrt 3 }{ 3 }) }{ (1 - \frac{ \sqrt3 }{ 3 })} \times \frac{ (1 - \frac{ \sqrt3 }{ 3 }) }{ (1 - \frac{ \sqrt3 }{ 3 }) } = \frac{ (1 + \frac{ \sqrt 3 }{ 3 })(1 - \frac{ \sqrt3 }{ 3 }) }{ (1 - \frac{ \sqrt3 }{ 3 })(1 - \frac{ \sqrt3 }{ 3 })}\]

Answer 11

\[\frac{ a^2 - b^2 }{ (a - b)^2}\]

Basically

Answer 12

what do

Answer 13

by the NUMERATOR

Answer 14

lmao

Answer 15

\[\frac{ (1 + \frac{ \sqrt 3 }{ 3 }) }{ (1 - \frac{ \sqrt3 }{ 3 })} \times \frac{ (1 + \frac{ \sqrt3 }{ 3 }) }{ (1 + \frac{ \sqrt3 }{ 3 }) } = \frac{ (1 + \frac{ \sqrt3 }{ 3 })(1 + \frac{ \sqrt3 }{ 3 }) }{ (1 - \frac{ \sqrt3 }{ 3 })(1 + \frac{ \sqrt3 }{ 3 }) } \]

Answer 16

oh

Answer 17

sec

Answer 18

\[\frac{ 1 + \frac{ 2 \sqrt 3 }{ 3 } + \frac{ 1 }{ 3 }}{ 1 - \frac{ 1 }{3 } }\]
\[\frac{ \frac{ 2 \sqrt 3 }{ 3 } + \frac{ 4 }{3 }}{ \frac{ 2 }{ 3 } }\]
\[\frac{ 4 + 2 \sqrt 3 }{ 3 } \times \frac{ 3 }{ 2 }\]
\[\frac{ 4 + 2 \sqrt3 }{ 2 } = 2 + \sqrt 3\]

Answer 19

This is a sign that I should get actual sleep tonight.

Answer 20

Thank you for the patience Inky

Answer 21

What I would have done:
\(\dfrac{1 + \dfrac{1}{\sqrt{3}}}{1 - \dfrac{1}{\sqrt{3}}}\)

Multiply top and bottom by \(\sqrt{3}\) to get
\(\dfrac{\sqrt{3} + 1}{\sqrt{3} - 1}\)

Then write an equivalent expression in the numerator like so:
\(\dfrac{\sqrt{3} - 1 + 2}{\sqrt{3} - 1}\)

Which then I can re-write as:
\(\dfrac{\sqrt{3} - 1}{\sqrt{3} - 1} + \dfrac{2}{\sqrt{3} - 1}\)

Which simplifies to
\(1 + \dfrac{2}{\sqrt{3} - 1}\)

Then simply multiply top and bottom by the conjugate of the denominator which is \(\sqrt{3} + 1\) to get:
\(1 + \dfrac{2}{\sqrt{3}-1} \cdot \dfrac{\sqrt{3}+1}{\sqrt{3}+1}\)

Which simplifies to
\(1 + \dfrac{2(\sqrt{3}+1)}{2}\)

Which further simplifies to
\(1 + \sqrt{3} + 1\)

Then just
\(2 + \sqrt{3}\)

Answer 22

Interesting approach. Thank you Hero

Answer 23

The overall approach is to get rid of the complex denominator first. Then multply by the conjugate. I probably could have eliminated a few steps had I did just that.