Question

show that each expression is rational
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HELP MEDALS WILL BE GIVEN

Answer 1

\[\frac{ 3 }{ 2 +\sqrt{2} } + \frac{ 3 }{ \sqrt{2} }\]

what does rational even mean :P

Answer 2

Rational means that it can be written as a fraction where the numerator and denominator are both integers.

Answer 3

mhm ok... but how can u prove it is rational or show it

Answer 4

multiply the top and bottom of the first term by 2-sqrt(2) and the top and bottom of the second term by sqrt(2). What happens?

Answer 5

\[\frac{ 3 }{ 2+\sqrt{2} }+\frac{ 3 }{ \sqrt{2} }\]
\[=\frac{ 3\sqrt{2} + 3(2+\sqrt{2}) }{ \sqrt{2}(2+\sqrt{2}) }\]
\[=\frac{ 6\sqrt{2}+6 }{ 2\sqrt{2}+2 }\]
\[=(\frac{ 6\sqrt{2}+6 }{ 2\sqrt{2}+2 })\times(\frac{ 2\sqrt{2} - 2 }{ 2\sqrt{2} - 2 })\]
\[=\frac{ (6\sqrt{2}+6)(2\sqrt{2} - 2) }{ (2\sqrt{2}+2)(2\sqrt{2} - 2) }\]
Expand the numerator as per usual, and then use your knowledge of completing the square in the denominator.
\[=\frac{ 12(2)-12\sqrt{2} + 12\sqrt{2} -12 }{ 4(2) - 4 }\]

Answer 6

All the terms with square root signs cancel out. Therefore you have a rationalised term.
\[=\frac{ 12 }{ 4 }\]

Answer 7

wow thanks so much