Rationalize the denominator 1/root 7+root 3-root2
Answers
Step-by-step explanation:
Here, the given expression is,
\frac{1}{\sqrt{7} +\sqrt{3} -\sqrt{2} }
7
+
3
−
2
1
Multiply and divide by √7 + √3 + √2,
=\frac{1}{\sqrt{7} +\sqrt{3} -\sqrt{2} }\times \frac{\sqrt{7} +\sqrt{3} +\sqrt{2}}{\sqrt{7} +\sqrt{3} +\sqrt{2}}=
7
+
3
−
2
1
×
7
+
3
+
2
7
+
3
+
2
=\frac{\sqrt{7} +\sqrt{3} +\sqrt{2}}{(\sqrt{7} +\sqrt{3})^2-(\sqrt{2})^2 }=
(
7
+
3
)
2
−(
2
)
2
7
+
3
+
2
=\frac{\sqrt{7} +\sqrt{3} +\sqrt{2}}{7+3+2\sqrt{21}-2}=
7+3+2
21
−2
7
+
3
+
2
=\frac{\sqrt{7} +\sqrt{3} +\sqrt{2}}{8+2\sqrt{21}}=
8+2
21
7
+
3
+
2
Multiply and divide by 8-2√21,
=\frac{\sqrt{7} +\sqrt{3} +\sqrt{2}}{8+2\sqrt{21}}\times \frac{8-2\sqrt{21}}{8-2\sqrt{21}}=
8+2
21
7
+
3
+
2
×
8−2
21
8−2
21
=\frac{8\sqrt{7} +8\sqrt{3} +9\sqrt{2}-2\sqrt{147} -2\sqrt{63} -2\sqrt{42}}{(8)^2-(2\sqrt{21})^2}=
(8)
2
−(2
21
)
2
8
7
+8
3
+9
2
−2
147
−2
63
−2
42
=\frac{8\sqrt{7} +8\sqrt{3} +9\sqrt{2}-14\sqrt{3} -6\sqrt{7} -2\sqrt{42}}{64-84}=
64−84
8
7
+8
3
+9
2
−14
3
−6
7
−2
42
=\frac{8\sqrt{7} +8\sqrt{3} +9\sqrt{2}-14\sqrt{3} -6\sqrt{7} -2\sqrt{42}}{-20}=
−20
8
7
+8
3
+9
2
−14
3
−6
7
−2
42
=\frac{2\sqrt{7} -6\sqrt{3} +9\sqrt{2}-2\sqrt{42}}{-20}=
−20
2
7
−6
3
+9
2
−2
42
=-\frac{1}{20}(2\sqrt{7}-6\sqrt{3}+8\sqrt{2}-2\sqrt{42})=−
20
1
(2
7
−6
3
+8
2
−2
42
)