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Simplify. Rationalize the denominator. \newline55+3\frac{5}{5 + \sqrt{3}}

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Full solution

Q. Simplify. Rationalize the denominator. \newline55+3\frac{5}{5 + \sqrt{3}}
  1. Find Conjugate: Select the conjugate of 5+35 + \sqrt{3}. The conjugate of a+ba + \sqrt{b} is a−ba - \sqrt{b}. So, the conjugate of 5+35 + \sqrt{3} is 5−35 - \sqrt{3}.
  2. Multiply by Conjugate: Multiply the original expression by 11 in the form of the conjugate over itself.\newlineTo rationalize the denominator, we multiply the numerator and the denominator by the conjugate of the denominator.\newline55+3\frac{5}{5 + \sqrt{3}} * 5−35−3\frac{5 - \sqrt{3}}{5 - \sqrt{3}}
  3. Simplify Numerator: Simplify the numerator.\newlineMultiply 55 by the conjugate (5−3)(5 - \sqrt{3}).\newline5×(5−3)=25−535 \times (5 - \sqrt{3}) = 25 - 5\sqrt{3}
  4. Simplify Denominator: Simplify the denominator.\newlineUse the difference of squares formula: (a+b)(a−b)=a2−b2(a + b)(a - b) = a^2 - b^2.\newline(5+3)(5−3)=52−(3)2=25−3(5 + \sqrt{3})(5 - \sqrt{3}) = 5^2 - (\sqrt{3})^2 = 25 - 3
  5. Calculate Denominator: Calculate the denominator. 25−3=2225 - 3 = 22
  6. Write Simplified Expression: Write the simplified expression. (25−53)/22(25 - 5\sqrt{3})/22 This fraction is already in simplest form.

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