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NECO 2008 Mathematics · Question 31

Rationalize the expression (7−3)(13−3)\frac{(7 - \sqrt{3})}{(13 - \sqrt{3})}

Options

  1. A.(44 + 33\sqrt{3}) ÷ 83
  2. B.(44 + 23\sqrt{3}) ÷ 83
  3. C.(44 - 33\sqrt{3}) ÷ 83
  4. D.(44 - 43\sqrt{3}) ÷ 83
  5. E.(44 - 23\sqrt{3}) ÷ 83

Answer: C. (44 - 33\sqrt{3}) ÷ 83

Explanation written with AI and checked against the answer key

Step 1: Analyze the question We need to remove the surd from the denominator. The denominator is 13 − √3, so we multiply the top and bottom by its conjugate, 13 + √3. Step 2: The concept (a − b)(a + b) = a² − b², which turns the denominator into a rational number. Step 3: Solution Numerator: (7 − √3)(13 + √3) = 91 + 7√3 − 13√3 − 3 = 88 − 6√3 Denominator: (13 − √3)(13 + √3) = 169 − 3 = 166 So the expression is: (88 − 6√3) / 166 Divide the top and bottom by 2: (44 − 3√3) / 83 Step 4: Answer Tip: When the denominator is of the form a − √b, always multiply by a + √b (the conjugate). After expanding, check whether everything can be divided by a common factor, as we did here by dividing by 2. That helps you match the options quickly.

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