Question

Find a and b if4 + 3 root 5 upon 4 + 3 root 5 = a+ b root 5

Answer 1

Step-by-step explanation:

a=−

29

61

and a=-\frac{24}{29}a=−

29

24

Step-by-step explanation:

Given : \frac{4+3\sqrt{5}}{4-3\sqrt{5}}=a+b\sqrt{5}

4−3

5

4+3

5

=a+b

5

To find : The value of a and b?

Solution :

Taking LHS,

\frac{4+3\sqrt{5}}{4-3\sqrt{5}}

4−3

5

4+3

5

Rationalize the expression,

=\frac{4+3\sqrt{5}}{4-3\sqrt{5}}\times\frac{4+3\sqrt{5}}{4+3\sqrt{5}}=

4−3

5

4+3

5

×

4+3

5

4+3

5

=\frac{(4+3\sqrt{5})^2}{(4)^2-(3\sqrt{5})^2}=

(4)

2

−(3

5

)

2

(4+3

5

)

2

=\frac{16+45+24\sqrt5}{16-45}=

16−45

16+45+24

5

=\frac{61+24\sqrt5}{-29}=

−29

61+24

5

=\frac{61}{-29}+\frac{24\sqrt5}{-29}=

−29

61

+

−29

24

5

On comparing with RHS,

a=-\frac{61}{29}a=−

29

61

and a=-\frac{24}{29}a=−

29

24