if √5+√3/2√5-3√3=a-b√15 find a&b
Answers
Answer:
Answer: The required values of a and b are
a=-\dfrac{19}{7},~~b=\dfrac{5}{7}.a=−
7
19
, b=
7
5
.
Step-by-step explanation: We are given to find the values of a and b from the following :
\dfrac{\sqrt5+\sqrt3}{2\sqrt5-3\sqrt3}=a-b\sqrt{15}~~~~~~~~~~~~~~~~~~~~~~~~~(i)
2
5
−3
3
5
+
3
=a−b
15
(i)
To find the required values of a and b, we need to rationalize the denominator on the L.H.S. of (i).
From equation (i), we have
\begin{gathered}\dfrac{\sqrt5+\sqrt3}{2\sqrt5-3\sqrt3}=a-b\sqrt{15}\\\\\\\Rightarrow \dfrac{(\sqrt5+\sqrt3)(2\sqrt5+3\sqrt3)}{(2\sqrt5-3\sqrt3)(2\sqrt5+3\sqrt3)}=a-b\sqrt{15}\\\\\\\Rightarrow \dfrac{2\times5+3\sqrt{3\times5}+2\sqrt{3\times5}+3\times3}{(2\sqrt5)^2-(3\sqrt3)^2}=a-b\sqrt{15}\\\\\\\Rightarrow \dfrac{19+5\sqrt{15}}{20-27}=a-b\sqrt{15}\\\\\\\Rightarrow -\dfrac{19}{7}-\dfrac{5}{7}\sqrt{15}=a-b\sqrt{15}.\end{gathered}
2
5
−3
3
5
+
3
=a−b
15
⇒
(2
5
−3
3
)(2
5
+3
3
)
(
5
+
3
)(2
5
+3
3
)
=a−b
15
⇒
(2
5
)
2
−(3
3
)
2
2×5+3
3×5
+2
3×5
+3×3
=a−b
15
⇒
20−27
19+5
15
=a−b
15
⇒−
7
19
−
7
5
15
=a−b
15
.
Comparing the corresponding coefficients in the above equation, we get
a=-\dfrac{19}{7},~~b=\dfrac{5}{7}.a=−
7
19
, b=
7
5
.
Thus, the required values of a and b are
a=-\dfrac{19}{7},~~b=\dfrac{5}{7}.a=−
7
19
, b=
7
5
.
hi dear
hopes its helps you
