Math, asked by lovelykhandelwal22, 2 months ago

4√2/√15-3√2 + 3√5/√10-√3 + 5√5/√6+√5​

Answers

Answered by mrravichandran988
1

Heya !!!

\begin{gathered} \frac{4 \sqrt{2} }{ \sqrt{15} - 3 \sqrt{2} } + \frac{3 \sqrt{5} }{ \sqrt{10} - \sqrt{3} } + \frac{5 \sqrt{3} }{ \sqrt{6} + \sqrt{5} } \\\end{gathered}

15

−3

2

4

2

+

10

3

3

5

+

6

+

5

5

3

On rationalizing the denominator we get,

\begin{gathered} = \frac{4 \sqrt{2} }{ \sqrt{15} - 3 \sqrt{2} } \times \frac{ \sqrt{15} + 3 \sqrt{2} }{ \sqrt{15} + 3 \sqrt{2} } + \frac{3 \sqrt{5} }{ \sqrt{10} - \sqrt{3} } \times \frac{ \sqrt{10} + \sqrt{3} }{ \sqrt{10} + \sqrt{3} } + \frac{5 \sqrt{3} }{ \sqrt{6} + \sqrt{5} } \times \frac{ \sqrt{6} - \sqrt{5} }{ \sqrt{6} - \sqrt{5} } \\ \\ = \frac{4 \sqrt{2} ( \sqrt{15} + 3 \sqrt{2} )}{ {( \sqrt{15}) }^{2} - {(3 \sqrt{2} )}^{2} } + \frac{3 \sqrt{5}( \sqrt{10} + \sqrt{3} )}{ {( \sqrt{10}) }^{2} - {( \sqrt{3} )}^{2} } + \frac{5 \sqrt{3} ( \sqrt{6} - \sqrt{5}) }{ {( \sqrt{6}) }^{2} - {( \sqrt{5}) }^{2} } \\ \\ = \frac{4 \sqrt{30} + 24 }{15 - 18} + \frac{3 \sqrt{50} + 3 \sqrt{15} }{10 - 3} + \frac{5 \sqrt{18} - 5 \sqrt{15} }{6 - 5} \\ \\ = \frac{4 \sqrt{30} + 24}{ - 3} + \frac{3 \sqrt{2 \times 5 \times 5} + 3 \sqrt{15} }{7} + 5 \sqrt{3 \times 3 \times 2} - 5 \sqrt{15} \\ \\ = \frac{ - 4 \sqrt{30} - 24}{3} + \frac{15 \sqrt{2} + 3 \sqrt{15} }{7} + 15 \sqrt{2} - 5 \sqrt{15} \\ \\ = \frac{ - 4 \sqrt{30} \times 7 - 24 \times 7 + 15 \sqrt{2} \times 3 + 3 \sqrt{15} \times 3 + 15 \sqrt{2} \times 21 - 5 \sqrt{15} \times 21}{21} \\ \\ = \frac{ - 28 \sqrt{30} - 168 + 45 \sqrt{2} + 9 \sqrt{15} + 315 \sqrt{2} - 105 \sqrt{15} }{21} \\ \\ = \frac{ - 28 \sqrt{30} - 168 + 360 \sqrt{2} - 96 \sqrt{15} }{21} \end{gathered}

=

15

−3

2

4

2

×

15

+3

2

15

+3

2

+

10

3

3

5

×

10

+

3

10

+

3

+

6

+

5

5

3

×

6

5

6

5

=

(

15

)

2

−(3

2

)

2

4

2

(

15

+3

2

)

+

(

10

)

2

−(

3

)

2

3

5

(

10

+

3

)

+

(

6

)

2

−(

5

)

2

5

3

(

6

5

)

=

15−18

4

30

+24

+

10−3

3

50

+3

15

+

6−5

5

18

−5

15

=

−3

4

30

+24

+

7

3

2×5×5

+3

15

+5

3×3×2

−5

15

=

3

−4

30

−24

+

7

15

2

+3

15

+15

2

−5

15

=

21

−4

30

×7−24×7+15

2

×3+3

15

×3+15

2

×21−5

15

×21

=

21

−28

30

−168+45

2

+9

15

+315

2

−105

15

=

21

−28

30

−168+360

2

−96

15

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