Rationalize the denominator \frac { 1 + \sqrt { 2 } } { \sqrt { 5 } + \sqrt { 3 } } + \frac { 1 - \sqrt { 2 } } { \sqrt { 5 } - \sqrt { 3 } } ​

Question

Rationalize the denominator


`\frac { 1 + \sqrt { 2 } } { \sqrt { 5 } + \sqrt { 3 } } + \frac { 1 - \sqrt { 2 } } { \sqrt { 5 } - \sqrt { 3 } }`



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Answer 1

√5 -√6

Explanation:

You want the simplified form of ...

`\frac { 1 + \sqrt { 2 } } { \sqrt { 5 } + \sqrt { 3 } } + \frac { 1 - \sqrt { 2 } } { \sqrt { 5 } - \sqrt { 3 } }`

Simplify

These fractions can be added in the usual way:

`\frac { 1 + \sqrt { 2 } } { \sqrt { 5 } + \sqrt { 3 } } + \frac { 1 - \sqrt { 2 } } { \sqrt { 5 } - \sqrt { 3 } }=\frac { (1 + \sqrt { 2 })(\sqrt { 5 } - \sqrt { 3 })+(1 -√(2)) (\sqrt { 5 } + \sqrt { 3 })} { (\sqrt { 5 } + \sqrt { 3 })(\sqrt { 5 } - \sqrt { 3 }) } \\\\\\=((√(5)-√(3)+√(10)-√(6))+(√(5)+√(3)-√(10)-√(6)))/(5-3)=(2(√(5)-√(6)))/(2)\\\\\\=\boxed{√(5)-√(6)}`

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Additional comment

You recognize that the common denominator will be the product of the factors of the difference of squares, so the denominator will be automatically rationalized by carrying out the addition.