Question

Conjugate/Rational Number?

Please include a detailed explanation so I can learn to do it by myself, Thank you!

Answer 1

1)
`(2)/(√(5) ) = (2 \cdot √(5) )/(5)`

2)
`-(5)/(√(3) ) = -(5 \cdot √(3) )/(3)`

3)
`(√(2) + √(5) )/(√(10) ) =(√(5) )/(5) + ( √(2) )/(2)`

4)
`(3 + √(2) )/(√(3) ) * (√(3) )/(√(3) ) = √(3) + (√(6) )/(3)`

5)
`(√(3) )/(√(5) + √(2) )= (√(15) - √(6) )/(3)`

Explanation:

The rationalization of the denominator of the surds are found as follows;

1)
`(2)/(√(5) )`

`(2)/(√(5) ) * (√(5) )/(√(5) ) = (2 \cdot √(5) )/(5)`

`(2)/(√(5) ) = (2 \cdot √(5) )/(5)`

2)
`-(5)/(√(3) )`

`-(5)/(√(3) ) * (√(3) )/(√(3) ) = -(5 \cdot √(3) )/(3)`

`-(5)/(√(3) ) = -(5 \cdot √(3) )/(3)`

3)
`(√(2) + √(5) )/(√(10) )`

`(√(2) + √(5) )/(√(10) ) * ( √(10) )/(√(10) ) = (√(20) + √(50) )/(10 ) = (2\cdot √(5) + 5 \cdot √(2) )/(10) = (√(5) )/(5) + ( √(2) )/(2)`

`(√(2) + √(5) )/(√(10) ) =(√(5) )/(5) + ( √(2) )/(2)`

4)
`(3 + √(2) )/(√(3) )`

`(3 + √(2) )/(√(3) ) * (√(3) )/(√(3) ) = (3 \cdot √(3)+√(6) )/(3 ) = √(3) + (√(6) )/(3)`

`(3 + √(2) )/(√(3) ) * (√(3) )/(√(3) ) = √(3) + (√(6) )/(3)`

5)
`(√(3) )/(√(5) + √(2) )`

`(√(3) )/(√(5) + √(2) ) = (√(5) - √(2) )/(√(5) - √(2) ) = (√(15) -√(6) )/(5 - 2) = (√(15) - √(6) )/(3)`

`(√(3) )/(√(5) + √(2) )= (√(15) - √(6) )/(3)`

6)
`(√(7) )/(√(3) - √(5) )`

`(√(7) )/(√(3) - √(5) ) * (√(3) + √(5))/(√(3) + √(5)) = \frac{√(21) + √(35)}{{3} + {5}} = (√(21) + √(35))/(8)`

`(√(7) )/(√(3) - √(5) ) * (√(3) + √(5))/(√(3) + √(5)) =(√(21) + √(35))/(8)`