Rationalize the denominator of
(i) a) 7 + 3√5
3 + √5
b ) 1
2 + √3
ii) Find three different irrational numbers between the rational numbers 6
7
and 9
11
.
Answers
Step-by-step explanation:
Rationalize the denominator of
(i) a) 7 + 3√5
3 + √5
b ) 1
2 + √3
ii) Find three different irrational numbers between the rational numbers 6
7
and 9
11
Step-by-step explanation:
(x+y)
2
=x
2
+y
2
+2xy
(x+y)(x−y)=x
2
−y
2
\begin{gathered} \frac{7 + 3 \sqrt{5} }{7 - 3 \sqrt{5} } \\ \end{gathered}
7−3
5
7+3
5
On rationalizing the denominator we get,
\begin{gathered} = \frac{7 + 3 \sqrt{5} }{7 - 3 \sqrt{5} } \times \frac{7 + 3 \sqrt{5} }{7 + 3 \sqrt{5} } \\ \\ = \frac{ {(7)}^{2} + {(3 \sqrt{5} )}^{2} + 2(7)(3 \sqrt{5} )}{ {(7)}^{2} - {(3 \sqrt{5}) }^{2} } \\ \\ = \frac{49 + 45 + 42 \sqrt{5} }{49 - 45} \\ \\ = \frac{94 + 42 \sqrt{5} }{4} \\ \\ = \frac{47 + 21 \sqrt{5} }{2} \\ \end{gathered}
=
7−3
5
7+3
5
×
7+3
5
7+3
5
=
(7)
2
−(3
5
)
2
(7)
2
+(3
5
)
2
+2(7)(3
5
)
=
49−45
49+45+42
5
=
4
94+42
5
=
2
47+21
5
frist answer