Rationalize the denominator and simplify:4 square root 5+3square root 2 divide 2square root 2-square root 5

Answer 1

( 4 √5 + 3 √2 ) / ( 2 √2 - √5 ) =

Multiply by ( 2 √2 + √5 ) / ( 2 √2 + √5 )

( 4 √5 + 3 √2 ) ∙ ( 2 √2 + √5 ) / ( 2 √2 - √5 ) ∙ ( 2 √2 + √5 ) =

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Remark:

( a - b ) ∙ ( a + b ) = a² - b²

so

( 2 √2 - √5 ) ∙ ( 2 √2 + √5 )= ( 2 √2 )² - √5²

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( 2 √2 ∙ 4 √5 + 2 √2 ∙ 3 √2 + √5 ∙ 4 √5 + √5 ∙ 3 √2 ) / [ ( 2 √2 )² - √5² ] =

( 8 √10 + 6 ∙ 2 + 4 ∙ 5 + 3 √10 ) / ( 4 ∙ 2 - 5 ) =

( 11 √10 + 12 + 20 ) / ( 8 - 5 ) =

( 11 √10 + 32 ) / 3