Question

1f * (sqrt(5) + sqrt(3)) / (sqrt(5) - sqrt(3)) = a + b * sqrt(15) then find the values of a and b.​

Answer 1

Appropriate Question :-

If $\rm{ \frac{ \sqrt{5} + \sqrt{3} }{ \sqrt{5} - \sqrt{3} } =a + b \sqrt{15} }$ then find the value of a and b.

Answer :-

• a = 4
• b = 1

Step by step explanation :-

Here, we cannot be directly solve the question we have first rationalize the denominator on LHS.

$\rm:\longmapsto{ \frac{ \sqrt{5} + \sqrt{3} }{ \sqrt{5} - \sqrt{3} } \times \frac{ \sqrt{5} + \sqrt{3} }{ \sqrt{5} + \sqrt{3} } } \\ \\ \rm:\longmapsto{ \frac{( \sqrt{5} + \sqrt{3} )( \sqrt{5} + \sqrt{3} )}{( \sqrt{5} - \sqrt{3} )( \sqrt{5} + \sqrt{3} )} }$

Using (a+b)² = a² + 2ab + b² in numerator and (a+b) (a-b) = a² - b² in the denominator.

$\rm:\longmapsto{ \frac{( \sqrt{5}) {}^{2} + 2( \sqrt{5} )( \sqrt{3} ) + ( \sqrt{3}) {}^{2} }{( \sqrt{5}) {}^{2} - ( \sqrt{3} ) {}^{2} } } \\ \\ \rm:\longmapsto{ \frac{5 + 2 \sqrt{15} + 3}{5 - 3} } \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\\rm:\longmapsto{ \frac{8 + 2 \sqrt{15} }{2} } \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:$

Taking 2 common in numerator and denominator.

$\rm:\longmapsto{ \frac{ \cancel{2}(4 + \sqrt{15} )}{ \cancel{2}} } \\ \\\bf:\longmapsto \red{4 + \sqrt {15} } \: \: \: \:$

Comparing LHS and RHS,

$\rm:\longmapsto{4 + \sqrt{15} = a + b \sqrt{15} }$

On comparing both sides we get,

• a = 4
• b = 1

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Answer 2

Appropriate Question :-

If \begin{gathered} \rm{ \frac{ \sqrt{5} + \sqrt{3} }{ \sqrt{5} - \sqrt{3} } =a + b \sqrt{15} } \\ \end{gathered}

5

−

3

5

+

3

=a+b

15

then find the value of a and b.

Answer :-

a = 4

b = 1

Step by step explanation :-

Here, we cannot be directly solve the question we have first rationalize the denominator on LHS.

\begin{gathered}\rm:\longmapsto{ \frac{ \sqrt{5} + \sqrt{3} }{ \sqrt{5} - \sqrt{3} } \times \frac{ \sqrt{5} + \sqrt{3} }{ \sqrt{5} + \sqrt{3} } } \\ \\ \rm:\longmapsto{ \frac{( \sqrt{5} + \sqrt{3} )( \sqrt{5} + \sqrt{3} )}{( \sqrt{5} - \sqrt{3} )( \sqrt{5} + \sqrt{3} )} }\end{gathered}

:⟼

5

−

3

5

+

3

×

5

+

3

5

+

3

:⟼

(

5

−

3

)(

5

+

3

)

(

5

+

3

)(

5

+

3

)

Using (a+b)² = a² + 2ab + b² in numerator and (a+b) (a-b) = a² - b² in the denominator.

\begin{gathered}\rm:\longmapsto{ \frac{( \sqrt{5}) {}^{2} + 2( \sqrt{5} )( \sqrt{3} ) + ( \sqrt{3}) {}^{2} }{( \sqrt{5}) {}^{2} - ( \sqrt{3} ) {}^{2} } } \\ \\ \rm:\longmapsto{ \frac{5 + 2 \sqrt{15} + 3}{5 - 3} } \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\\rm:\longmapsto{ \frac{8 + 2 \sqrt{15} }{2} } \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \end{gathered}

:⟼

(

5

)

2

−(

3

)

2

(

5

)

2

+2(

5

)(

3

)+(

3

)

2

:⟼

5−3

5+2

15

+3

:⟼

2

8+2

15

Taking 2 common in numerator and denominator.

\begin{gathered}\rm:\longmapsto{ \frac{ \cancel{2}(4 + \sqrt{15} )}{ \cancel{2}} } \\ \\\bf:\longmapsto \red{4 + \sqrt {15} } \: \: \: \: \end{gathered}

:⟼

2

2

(4+

15

)

:⟼4+

15

Comparing LHS and RHS,

\rm:\longmapsto{4 + \sqrt{15} = a + b \sqrt{15} }:⟼4+

15

=a+b

15

On comparing both sides we get,

a = 4

b = 1

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