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Simplify. Rationalize the denominator. \newline107+5\frac{10}{7 + \sqrt{5}}

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Full solution

Q. Simplify. Rationalize the denominator. \newline107+5\frac{10}{7 + \sqrt{5}}
  1. Select Conjugate: Select the conjugate of 7+57 + \sqrt{5}.\newlineConjugate of a+ba + \sqrt{b}: a−ba - \sqrt{b}\newlineConjugate of 7+57 + \sqrt{5}: 7−57 - \sqrt{5}
  2. Multiply by Conjugate: Multiply the original expression by the conjugate over itself to rationalize the denominator.\newline(107+5)⋅7−57−5(\frac{10}{7 + \sqrt{5}}) \cdot \frac{7 - \sqrt{5}}{7 - \sqrt{5}}
  3. Simplify Numerator: Simplify the numerator by distributing the multiplication.\newline10×(7−5)=70−10×510 \times (7 - \sqrt{5}) = 70 - 10 \times \sqrt{5}
  4. Simplify Denominator: Simplify the denominator by using the difference of squares formula.\newline(7+5)×(7−5)=72−(5)2=49−5=44(7 + \sqrt{5}) \times (7 - \sqrt{5}) = 7^2 - (\sqrt{5})^2 = 49 - 5 = 44
  5. Write with Rationalized Denominator: Write the simplified expression with the rationalized denominator. (70−10⋅5)/44(70 - 10 \cdot \sqrt{5}) / 44
  6. Simplify Numerator and Denominator: Simplify the expression by dividing both terms in the numerator by the denominator. 7044−(10⋅5)44\frac{70}{44} - \frac{(10 \cdot \sqrt{5})}{44}
  7. Reduce Fractions: Reduce the fractions to their simplest form if possible.\newline7044=3522\frac{70}{44} = \frac{35}{22} and 10⋅544=5⋅522\frac{10 \cdot \sqrt{5}}{44} = \frac{5 \cdot \sqrt{5}}{22}
  8. Write Final Expression: Write the final simplified expression. 3522−5⋅522\frac{35}{22} - \frac{5 \cdot \sqrt{5}}{22}

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