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Rationalize the first term, sqrt(2)sqrt(2)-sqrt(3).

Mathematics

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Answer

-5

Step-by-step solution

4 steps

To evaluate the expression 22−3−33+2\frac{\sqrt{2}}{\sqrt{2}-\sqrt{3}} - \frac{\sqrt{3}}{\sqrt{3}+\sqrt{2}} by rationalizing the denominator, we will rationalize each fraction separately.

  1. Rationalize the first term, 22−3\frac{\sqrt{2}}{\sqrt{2}-\sqrt{3}}.

    Multiply the numerator and denominator by the conjugate of the denominator, which is 2+3\sqrt{2}+\sqrt{3}.

    22−3×2+32+3=2(2+3)(2)2−(3)2\frac{\sqrt{2}}{\sqrt{2}-\sqrt{3}} \times \frac{\sqrt{2}+\sqrt{3}}{\sqrt{2}+\sqrt{3}} = \frac{\sqrt{2}(\sqrt{2}+\sqrt{3})}{(\sqrt{2})^2 - (\sqrt{3})^2} =2+62−3=2+6−1=−2−6= \frac{2 + \sqrt{6}}{2 - 3} = \frac{2 + \sqrt{6}}{-1} = -2 - \sqrt{6}
  2. Rationalize the second term, 33+2\frac{\sqrt{3}}{\sqrt{3}+\sqrt{2}}.

    Multiply the numerator and denominator by the conjugate of the denominator, which is 3−2\sqrt{3}-\sqrt{2}.

    33+2×3−23−2=3(3−2)(3)2−(2)2\frac{\sqrt{3}}{\sqrt{3}+\sqrt{2}} \times \frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}-\sqrt{2}} = \frac{\sqrt{3}(\sqrt{3}-\sqrt{2})}{(\sqrt{3})^2 - (\sqrt{2})^2} =3−63−2=3−61=3−6= \frac{3 - \sqrt{6}}{3 - 2} = \frac{3 - \sqrt{6}}{1} = 3 - \sqrt{6}
  3. Subtract the rationalized second term from the rationalized first term.

    (−2−6)−(3−6)(-2 - \sqrt{6}) - (3 - \sqrt{6}) =−2−6−3+6= -2 - \sqrt{6} - 3 + \sqrt{6}
  4. Combine like terms.

    (−2−3)+(−6+6)(-2 - 3) + (-\sqrt{6} + \sqrt{6}) =−5+0=−5= -5 + 0 = -5

    The final answer is −5-5.

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