Rationalize the denominator:
3−223+22\dfrac{3 - 2\sqrt{2}}{3 + 2\sqrt{2}}3+223−22
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Rationalizing the denominator,
⇒3−223+22×3−223−22⇒(3−22)2(3)2−(22)2⇒(3)2+(22)2−2×3×229−8⇒(9+8−122)1⇒(17−122)\Rightarrow \dfrac{3 - 2\sqrt{2}}{3 + 2\sqrt{2}} \times \dfrac{3 - 2\sqrt{2}}{3 - 2\sqrt{2}} \\[1em] \Rightarrow \dfrac{(3 - 2\sqrt{2})^2} {(3)^2 - ( 2\sqrt{2})^2} \\[1em] \Rightarrow \dfrac{(3)^2 + ( 2\sqrt{2})^2 -2 \times 3 \times 2\sqrt{2} } {9-8} \\[1em] \Rightarrow \dfrac{(9 + 8 - 12\sqrt{2})} {1} \\[1em] \Rightarrow (17 - 12\sqrt{2}) \\[1em]⇒3+223−22×3−223−22⇒(3)2−(22)2(3−22)2⇒9−8(3)2+(22)2−2×3×22⇒1(9+8−122)⇒(17−122)
Hence, on rationalizing 3−223+22=(17−122)\dfrac{3 - 2\sqrt{2}}{3 + 2\sqrt{2}} = (17 - 12\sqrt{2})3+223−22=(17−122).
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