5+2root3 by 7+2root3 = a+broot3
Answers
Answer:
\large{\underline{\boxed{\sf a = \frac{29}{46} \: and \: b = - \frac{9}{46}}}}
a=
46
29
andb=−
46
9
Step-by-step explanation:
\begin{gathered} \sf \implies \frac{5 + 2 \sqrt{3} }{7 + \sqrt{3} } = a - \sqrt{3} b \\ \\ \sf \implies \frac{5 + 2 \sqrt{3} }{7 + \sqrt{3} } \times \frac{7 - \sqrt{3} }{7 - \sqrt{3} } \\ \\ \sf \implies \frac{(5 + 2 \sqrt{3})(7 - \sqrt{3})}{(7)^{2} - { (\sqrt{3} )}^{2} } \\ \\ \sf \implies \frac{35 - 5 \sqrt{3} + 14 \sqrt{3} - 6 }{49 - 3} \\ \\ \sf \implies \frac{29 + 9 \sqrt{3} }{46} \\ \\ \sf \implies \frac{29}{46} + \frac{9 }{46} \sqrt{3} \end{gathered}
⟹
7+
3
5+2
3
=a−
3
b
⟹
7+
3
5+2
3
×
7−
3
7−
3
⟹
(7)
2
−(
3
)
2
(5+2
3
)(7−
3
)
⟹
49−3
35−5
3
+14
3
−6
⟹
46
29+9
3
⟹
46
29
+
46
9
3
On comparing it with a - √3 b,
\boxed{ \bf a = \frac{29}{46} \: and \: b = - \frac{9}{46} }
a=
46
29
andb=−
46
9
_______________________
\large{\underline{\underline{\mathfrak{\heartsuit \: Extra \: Information: \: \heartsuit}}}}
♡ExtraInformation:♡
To rationalise the denominator, just multiply the whole fraction by its rationalising factor.
Example -
\sf \frac{1}{2 - \sqrt{3}}
2−
3
1
Here, the rationalising factor of 2 - √3 is 2 + √3.