Math, asked by nigel82, 6 months ago

If (√2 + √3)/(3√2 - 2√3) = a - b√6, then find the value of a and b.

Answers

Answered by MaIeficent
2

Step-by-step explanation:

Given:-

  • \sf \dfrac{ \sqrt{2} +  \sqrt{3}  }{3 \sqrt{2} - 2 \sqrt{3}  }  = a - b \sqrt{6}

To Find:-

  • Find the value of a and b

Solution:-

\sf \longrightarrow \dfrac{ \sqrt{2} +  \sqrt{3}  }{3 \sqrt{2} - 2 \sqrt{3}  }  = a - b \sqrt{6}

By rationalizing the denominator:-

\sf \longrightarrow \dfrac{ \sqrt{2} +  \sqrt{3}  }{3 \sqrt{2} - 2 \sqrt{3}  } \times  \dfrac{3 \sqrt{2}  + 2 \sqrt{3} }{3 \sqrt{2}  +  2\sqrt{3} }   = a - b \sqrt{6}

\sf \longrightarrow  \dfrac{( \sqrt{2}  +  \sqrt{3}) \times ( 3 \sqrt{2}  + 2 \sqrt{3}) }{(3 \sqrt{2}  - 2 \sqrt{3} ) \times (3 \sqrt{2}  +  2\sqrt{3}) }   = a - b \sqrt{6}

\sf \longrightarrow  \dfrac{\sqrt{2} (3 \sqrt{2} ) +  \sqrt{2}  (2 \sqrt{3} )  +  \sqrt{3}( 3 \sqrt{2} ) +  \sqrt{3}( 2 \sqrt{3}) }{(3 \sqrt{2} ) ^{2}  - (2 \sqrt{3} )^{2}  }   = a - b \sqrt{6}

\sf \longrightarrow  \dfrac{6 + 2 \sqrt{6} + 3 \sqrt{6}  +  6 }{18  - 12  }   = a - b \sqrt{6}

\sf \longrightarrow  \dfrac{6 +6 + 2 \sqrt{6}  + 3 \sqrt{6}    }{6  }   = a - b \sqrt{6}

\sf \longrightarrow  \dfrac{12 + 5\sqrt{6} }{6  } = a - b \sqrt{6}

\sf \longrightarrow \:  \dfrac{12}{6} +  \dfrac{ 5\sqrt{6} }{6  } = a - b \sqrt{6}

\sf \longrightarrow 2+  \dfrac{ 5\sqrt{6} }{6  } = a - b \sqrt{6}

\sf Comparing \: \:   2+  \dfrac{ 5\sqrt{6} }{6  }   \: \: with \:  \: a - b \sqrt{6}

 \dashrightarrow \underline{\boxed{ \bf a = 2   \: , \: b =   \frac{ - 5}{6}}  }

Answered by llAbdulkadarll
3

Data :

(√2 + √3)/(3√2 - 2√3) = a - b√6

Solving :

= ( √2 + √3 / 3 √2 - 2 √3 ) × ( 3 √2 + 2 √3 / 3 √2 + z √3 ) = a - b √6

= [ √2 ( 3 √2 ) + √2 ( 2 √3 ) + √3 ( 3 √2 ) + √3 ( 2 √3 ) ] / [ ( 3 √2 )^2 - ( 2 √3 )^2 ] = a - b √6

= ( 6 + 2 √6 + 3 √6 + 6 ) / ( 18 - 12 ) = a - b √6

= ( 6 + 6 + 2 √6 + 3 √6 ) / 6 = a - b √6

= ( 12 + 5 √6 ) / 6 = a - b √6

= 2 + 5 √6 / 6 = a - b √6

= Form = a - b √6

So, a = 2

a = 2 b = -5 / 6

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