solve the question please question is if a= root 5 - 3 / root 5 + root 3 and B = root 5 + root 3 / root 5 - root 3 find a + b + ab
Answers
Answer:
Given :-
\sf \dfrac{ \sqrt{5} + \sqrt{3} }{ \sqrt{5} - \sqrt{3} } = a + b \sqrt{15}
5
−
3
5
+
3
=a+b
15
Required to find :-
Values of " a " and " b "
Identities used :-
( x + y ) ( x + y ) = ( x + y )²
( x + y ) ( x - y ) = x² - y²
( x + y )² = x² + 2xy + y²
Solution :-
Given information :-
\rm \dfrac{ \sqrt{5} + \sqrt{3} }{ \sqrt{5} + \sqrt{3} } = a + b \sqrt{5}
5
+
3
5
+
3
=a+b
5
we need to find the values of ' a ' and ' b '
So,
Consider the LHS part
\tt \dfrac{ \sqrt{5} + \sqrt{3} }{ \sqrt{5} - \sqrt{3} }
5
−
3
5
+
3
Here,
We need to rationalize the denominator !
So,
Rationalising factor of √5 - √3 = √5 + √3
Hence,
Multiply both numerator and denominator with that factor
So,
\tt \dfrac{ \sqrt{5} + \sqrt{3} }{ \sqrt{5} - \sqrt{3} } \times \dfrac{ \sqrt{5} + \sqrt{3} }{ \sqrt{5} + \sqrt{3} }
5
−
3
5
+
3
×
5
+
3
5
+
3
Here we need to use some algebraic Identities
They are ,
1. ( x + y ) ( x + y ) = ( x + y )²
2. ( x + y ) ( x - y ) = x² - y²
3. ( x + y )² = x² + 2xy + y²
So,
Using 1 and 2 we get ;
\rm \dfrac{( \sqrt{5} + \sqrt{3} {)}^{2} }{( \sqrt{5} {)}^{2} - ( \sqrt{3} {)}^{2} }
(
5
)
2
−(
3
)
2
(
5
+
3
)
2
Using the 3rd identity expand the numerator
\tt \dfrac{( \sqrt{5} {)}^{2} + ( \sqrt{3} {)}^{2} + 2( \sqrt{5} )( \sqrt{3} ) }{5 - 3}
5−3
(
5
)
2
+(
3
)
2
+2(
5
)(
3
)
This implies,
\tt \dfrac{5 + 3 + 2 \sqrt{15} }{2}
2
5+3+2
15
\begin{gathered} \tt \dfrac{ 8 + 2 \sqrt{15} }{2} \\ \\ \tt \dfrac{2 \: ( \: 4 + \sqrt{15} \: ) }{2} \end{gathered}
2
8+2
15
2
2(4+
15
)
2 gets cancelled in both numerator and denominator
So,
we are left with ;
\tt 4 + \sqrt{15}4+
15
Now,
Compare the LHS and RHS parts
\rm 4 + \sqrt{15} = a + b \sqrt{15}4+
15
=a+b
15
From the above comparison we can conclude that the LHS in the form of RHS
So,
Equal the values on both sides
Hence,
a = 4
b = 1
Therefore,
Values of " a " and " b " are 4 & 1
Step-by-step explanation:
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Answer: