Question

Rationalise the denominator of 5/√(3-√5)
Pls send the answer by today​

Answer 1

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To rationalise:-

`\frac{5}{ \sqrt{3 - √(5) } }`

There is a formula in math if there is root in denominator

for example

`(1)/( √(a - b) )`

we can say rationalize by multiplying √(a-b) in numerator and denominator both

`(1)/( √(a - b) ) * ( √(a - b) )/( √(a - b) ) \\ ( √(a - b) )/(a - b)`

In here

`\frac{5}{ \sqrt{3 - √(5) } } * \frac{ \sqrt{3 - √(5) } }{ \sqrt{3 - { √(5) } } } \\ \frac{5( \sqrt{3 - √(5) }) }{3 - √(5) }`

but still here is root to remove this we have to multiply

3 + √5 in numerator and denominator.

`\frac{5( \sqrt{3 - √(5) }) }{3 - √(5) } * (3 + √(5) )/(3 + √(5) ) \\ \frac{5( \sqrt{3 - √(5) })(3 + √(5) ) }{ {3}^(2) - { (√(5) )}^(2) } \\ \frac{5( \sqrt{3 - √(5) })(3 + √(5)) }{9 - 5} \\ \frac{5( \sqrt{3 - √(5) } )(3 + √(5)) }{4}`

Answer 2

`\frac{5(3+√(5))\sqrt{3-√(5)}}{4}`

`\textsf{or}\quad \frac{5\sqrt{3+√(5)}}{2}`

Explanation:

`\textsf{Given expression}:\frac{5}{\sqrt{3-√(5)}}`

Method 1

`\textsf{Multiply by the conjugate}\quad \frac{\sqrt{3-√(5)}}{\sqrt{3-√(5)}}:`

`\implies \frac{5}{\sqrt{3-√(5)}} * \frac{\sqrt{3-√(5)}}{\sqrt{3-√(5)}}=\frac{5\sqrt{3-√(5)}}{(\sqrt{3-√(5)})(\sqrt{3-√(5)})}`

Simplify the denominator using the radical rule
`√(a) √(a) =a`:

`\implies (\sqrt{3-√(5)})(\sqrt{3-√(5)})=3-√(5)`

Therefore:

`\implies \frac{5\sqrt{3-√(5)}}{(\sqrt{3-√(5)})(\sqrt{3-√(5)})}= \frac{5\sqrt{3-√(5)}}{3-√(5)}`

`\textsf{Multiply by the conjugate}\quad (3+√(5))/(3+√(5)):`

`\implies \frac{5\sqrt{3-√(5)}}{3-√(5)} * (3+√(5))/(3+√(5))=\frac{5\sqrt{3-√(5)}(3+√(5))}{(3-√(5))(3+√(5))}`

Simplify the denominator:

`\implies (3-√(5))(3+√(5))=9+3√(5)-3√(5)-5=4`

Therefore:

`\implies \frac{5(3+√(5))\sqrt{3-√(5)}}{4}`

Method 2

`\textsf{Multiply by the conjugate}\quad \frac{\sqrt{3+√(5)}}{\sqrt{3+√(5)}}:`

`\implies \frac{5}{\sqrt{3-√(5)}} * \frac{\sqrt{3+√(5)}}{\sqrt{3+√(5)}}=\frac{5\sqrt{3+√(5)}}{(\sqrt{3-√(5)})(\sqrt{3+√(5)})}`

Simplify the denominator using the radical rule
`√(a) √(b) =√(ab)`:

`\implies (\sqrt{3-√(5)})(\sqrt{3+√(5)})=\sqrt{(3-√(5))(3+√(5))`

`\implies\sqrt{(3-√(5))(3+√(5))}=√(9-5)=√(4)=2`

Therefore:

`\implies \frac{5\sqrt{3+√(5)}}{2}`