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Simplify. Rationalize the denominator.\newline99−2\frac{9}{9 - \sqrt{2}}

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Full solution

Q. Simplify. Rationalize the denominator.\newline99−2\frac{9}{9 - \sqrt{2}}
  1. Identify Conjugate: Identify the conjugate of the denominator.\newlineThe conjugate of a−ba - \sqrt{b} is a+ba + \sqrt{b}. Therefore, the conjugate of 9−29 - \sqrt{2} is 9+29 + \sqrt{2}.
  2. Multiply by Conjugate: Multiply the numerator and the denominator by the conjugate of the denominator.\newlineTo rationalize the denominator, we multiply the expression by (9+2)/(9+2)(9 + \sqrt{2})/(9 + \sqrt{2}).\newline(9/(9−2))×((9+2)/(9+2))(9/(9 - \sqrt{2})) \times ((9 + \sqrt{2})/(9 + \sqrt{2}))
  3. Apply Distributive Property: Apply the distributive property to the numerator.\newlineMultiply 99 by each term in the conjugate (9+2)(9 + \sqrt{2}).\newline9×9+9×2=81+929 \times 9 + 9 \times \sqrt{2} = 81 + 9\sqrt{2}
  4. Apply Difference of Squares: Apply the difference of squares formula to the denominator.\newline(9−2)×(9+2)=92−(2)2=81−2(9 - \sqrt{2}) \times (9 + \sqrt{2}) = 9^2 - (\sqrt{2})^2 = 81 - 2
  5. Simplify Denominator: Simplify the denominator. 81−2=7981 - 2 = 79
  6. Write Simplified Expression: Write the simplified expression.\newlineThe rationalized expression is (81+92)/79(81 + 9\sqrt{2})/79.

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