find a and b if 3+ root 6 / root 3 + root 2 =a + b root 3
Answers
Step-by-step explanation:
Here is the answer you were looking for:
\begin{gathered}\frac{ \sqrt{3} + \sqrt{2} }{ \sqrt{3} - \sqrt{2} } = a + b \sqrt{6} \\\end{gathered}
3
−
2
3
+
2
=a+b
6
On rationalizing the denominator we get,
\begin{gathered}= \frac{ \sqrt{3} + \sqrt{2} }{ \sqrt{3} - \sqrt{2} } \times \frac{ \sqrt{3} + \sqrt{2} }{ \sqrt{3} + \sqrt{2} } \\ \\\end{gathered}
=
3
−
2
3
+
2
×
3
+
2
3
+
2
Using the identity :
\begin{gathered}{(a + b)}^{2} = {a}^{2} + {b}^{2} + 2ab \\ (a + b)(a - b) = {a}^{2} - {b}^{2}\end{gathered}
(a+b)
2
=a
2
+b
2
+2ab
(a+b)(a−b)=a
2
−b
2
\begin{gathered}= \frac{ {( \sqrt{3} })^{2} + {( \sqrt{2} )}^{2} + 2( \sqrt{3} )( \sqrt{2} )}{ {( \sqrt{3} )}^{2} - {( \sqrt{2} )}^{2} } \\ \\ = \frac{3 + 2 + 2 \sqrt{6} }{3 - 2} \\ \\ 5 + 2 \sqrt{6} = a + b \sqrt{6} \\ \\ a = 5 \: : \: b = 2\end{gathered}
=
(
3
)
2
−(
2
)
2
(
3
)
2
+(
2
)
2
+2(
3
)(
2
)
=
3−2
3+2+2
6
5+2
6
=a+b
6
a=5:b=2
Hope this helps!!