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Simplify. Rationalize the denominator. \newline7−7+5\frac{7}{-7 + \sqrt{5}}

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Full solution

Q. Simplify. Rationalize the denominator. \newline7−7+5\frac{7}{-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.\newline7−7+5×−7−5−7−5\frac{7}{-7 + \sqrt{5}} \times \frac{-7 - \sqrt{5}}{-7 - \sqrt{5}}
  3. Simplify Numerator: Simplify the numerator by distributing the 77 across the conjugate.7×(−7−5)7 \times (-7 - \sqrt{5})=7×(−7)+7×(−5)= 7 \times (-7) + 7 \times (-\sqrt{5})=−49−75= -49 - 7\sqrt{5}
  4. Simplify Denominator: Simplify the denominator by using the difference of squares formula.\newline(−7+5)∗(−7−5)(-7 + \sqrt{5}) * (-7 - \sqrt{5})\newline=(−7)2−(5)2= (-7)^2 - (\sqrt{5})^2\newline=49−5= 49 - 5\newline=44= 44
  5. Write Simplified Expression: Write the simplified expression with the rationalized denominator.\newline(−49−75)/44(-49 - 7\sqrt{5}) / 44\newlineThis fraction is already in simplest form.

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