Question

If 'a' and 'b' are rational number and 3+√5 / 3-√5 = a+b√5, find the value of 'a' and 'b' ?

Answer 1

Hey

Here is your answer,

3+√5 / 3-√5 =a+b√5

=3+√5/3-√5 x 3+√5/3+√5
=(3+√5)^2 / (3)^2 - (√5)^2
=9+5+6√5 / 9-5
=14+6√5 / 4
=2(7+3√5) / 4
=7+3√5/2

a=7/2
b=3/2

Hope it helps you!

Answer 2

Given,

⇒ ( 3 + √5 ) ÷ ( 3 - √5 ) = a + b√5

By multiplying the numerator and denominator of L.H.S by ( 3 + √5 ).

⇒ ( 3 + √5 )(3 + √5 ) ÷ ( 3 - √5 ) ( 3 + √5 ) = a + b√5

⇒ ( 3 + √5 )² ÷ { (3)² - (√5)² } = a + b√5

⇒ { 3² + (√5)² + 2×3×√5} ÷ ( 9 - 5 ) = a + b√5

⇒ ( 9 + 5 + 6√5 ) ÷ 4 = a + b√5

⇒ ( 14 + 6√5 ) ÷ 4 = a + b√5

⇒ 2 ( 7 + 3√5 ) ÷ 4 = a + b√5

⇒ ( 7 + 3√5 ) ÷ 2 = a + b√5

⇒ { ( 7/2 ) + ( 3√5 / 2 ) } = a + b√5

By comparing the coefficients,

⇒ a = 7/2 = 3.5

⇒ b√5 = 3√5/2

⇒ b = 3/2

∴ b = 1.5


Hence, a = 3.5 ( 7/2 ) and b = 1.5 ( 3/2 ).