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Simplify. Rationalize the denominator.\newline3−7−5\frac{3}{-7 - \sqrt{5}}

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Full solution

Q. Simplify. Rationalize the denominator.\newline3−7−5\frac{3}{-7 - \sqrt{5}}
  1. Select Conjugate: Select the conjugate of −7−5-7 - \sqrt{5}.\newlineConjugate of a−ba - \sqrt{b}: a+ba + \sqrt{b}\newlineConjugate of −7−5-7 - \sqrt{5}: −7+5-7 + \sqrt{5}
  2. Multiply by Conjugate: Multiply the numerator and the denominator by the conjugate of the denominator to rationalize it.\newline(3)/(−7−5)∗(−7+5)/(−7+5)(3) / (-7 - \sqrt{5}) * (-7 + \sqrt{5}) / (-7 + \sqrt{5})
  3. Apply Distributive Property: Apply the distributive property to multiply the numerators and the denominators.\newlineNumerator: 3×(−7+5)=−21+353 \times (-7 + \sqrt{5}) = -21 + 3\sqrt{5}\newlineDenominator: (−7−5)×(−7+5)=(−7)2−(5)2(-7 - \sqrt{5}) \times (-7 + \sqrt{5}) = (-7)^2 - (\sqrt{5})^2
  4. Simplify Denominator: Simplify the denominator.\newline(−7)2−(5)2=49−5=44(-7)^2 - (\sqrt{5})^2 = 49 - 5 = 44
  5. Write Simplified Expression: Write the simplified expression. −21+3544\frac{-21 + 3\sqrt{5}}{44}

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