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Bentuk rasional dari (2-√3)/(2+√3) adalah .....

Mathematics

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Answer

A. 7−43\text{A. } 7 - 4\sqrt{3}

Step-by-step solution

6 steps

  1. Identify the expression to be rationalized.

    The expression is 2−32+3\frac{2-\sqrt{3}}{2+\sqrt{3}}. To rationalize this expression, we need to eliminate the square root from the denominator.

  2. Multiply the numerator and denominator by the conjugate of the denominator.

    The denominator is 2+32+\sqrt{3}. Its conjugate is 2−32-\sqrt{3}.

    2−32+3×2−32−3\frac{2-\sqrt{3}}{2+\sqrt{3}} \times \frac{2-\sqrt{3}}{2-\sqrt{3}}

  3. Expand the numerator.

    The numerator becomes (2−3)(2−3)=(2−3)2(2-\sqrt{3})(2-\sqrt{3}) = (2-\sqrt{3})^2.
    Using the formula (a−b)2=a2−2ab+b2(a-b)^2 = a^2 - 2ab + b^2:

    (2−3)2=22−2(2)(3)+(3)2(2-\sqrt{3})^2 = 2^2 - 2(2)(\sqrt{3}) + (\sqrt{3})^2 =4−43+3= 4 - 4\sqrt{3} + 3 =7−43= 7 - 4\sqrt{3}

  4. Expand the denominator.

    The denominator becomes (2+3)(2−3)(2+\sqrt{3})(2-\sqrt{3}).
    Using the formula (a+b)(a−b)=a2−b2(a+b)(a-b) = a^2 - b^2:

    (2+3)(2−3)=22−(3)2(2+\sqrt{3})(2-\sqrt{3}) = 2^2 - (\sqrt{3})^2 =4−3= 4 - 3 =1= 1

  5. Combine the simplified numerator and denominator.

    7−431=7−43\frac{7 - 4\sqrt{3}}{1} = 7 - 4\sqrt{3}

  6. Compare the result with the given options.

    The result 7−437 - 4\sqrt{3} matches option A.

    The final answer is A. 7−43\text{A. } 7 - 4\sqrt{3}.

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