Question

rationalize the denominator and simplify 1 + root 2 upon 3 minus 2 root 2

Answer 1

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Answer 2

The given question is we have to rationalize and simplify the denominator.

The given expression is

$\frac{1 + \sqrt{2} }{3 - 2 \sqrt{2} }$

we have to rationalize the above expression, multiply with the conjugate term of the denominator

the conjugate term of

$3 - 2 \sqrt{2 } = 3 + 2 \sqrt{2}$

$\frac{1 + \sqrt{2} }{3 - 2 \sqrt{2} } \times \frac{3 + 2 \sqrt{2} }{3 + 2 \sqrt{2} } \\ \frac{1(3 + 2 \sqrt{2}) + \sqrt{2} (3 + 2 \sqrt{2} ) }{ ({3}^{2}) - ( { 2\sqrt{2} )}^{2} }$

the LCM is

$(3 - 2 \sqrt{2} )(3 + 2 \sqrt{2} )$

The next step is

$\frac{3 + 2 \sqrt{2} + 3 \sqrt{2} + 2 \times 2}{9 - 8}$

$\frac{ 3 + 5 \sqrt{2} + 4}{9 - 8}$

Adding the like terms we get the above expression.

$\frac{7 + 5 \sqrt{2} }{1}$

The final expression is

$7 + 5 \sqrt{2}$

Therefore the simplified form of the given expression is founded as

$7 + 5 \sqrt{2}$

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