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Simplify (6-4sqrt3)/(6+4sqrt3) by rationalizing the denominator.

Simplify 6−436+43 \frac{6-4 \sqrt{3}}{6+4 \sqrt{3}} by rationalizing the denominator.

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Full solution

Q. Simplify 6−436+43 \frac{6-4 \sqrt{3}}{6+4 \sqrt{3}} by rationalizing the denominator.
  1. Find Conjugate: First, find the conjugate of the denominator, which is 6−436 - 4\sqrt{3}.
  2. Multiply by Conjugate: Now, multiply the numerator and the denominator by the conjugate.\newline(6−43)/(6+43)×(6−43)/(6−43)(6-4\sqrt{3})/(6+4\sqrt{3}) \times (6-4\sqrt{3})/(6-4\sqrt{3})
  3. Expand Numerator: Expand the numerator: (6−43)(6−43)(6-4\sqrt{3})(6-4\sqrt{3}) = 6×6−6×43−43×6+43×436\times6 - 6\times4\sqrt{3} - 4\sqrt{3}\times6 + 4\sqrt{3}\times4\sqrt{3} = 36−243−243+4836 - 24\sqrt{3} - 24\sqrt{3} + 48 = 36+48−243−24336 + 48 - 24\sqrt{3} - 24\sqrt{3} = 84−48384 - 48\sqrt{3}
  4. Expand Denominator: Expand the denominator: (6+43)(6−43)(6+4\sqrt{3})(6-4\sqrt{3}) = 6×6−6×43+43×6−43×436\times 6 - 6\times 4\sqrt{3} + 4\sqrt{3}\times 6 - 4\sqrt{3}\times 4\sqrt{3} = 36−243+243−4836 - 24\sqrt{3} + 24\sqrt{3} - 48 = 36−4836 - 48 = −12-12

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