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Simplify. Rationalize the denominator.\newline95+5\frac{9}{5 + \sqrt{5}}

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Full solution

Q. Simplify. Rationalize the denominator.\newline95+5\frac{9}{5 + \sqrt{5}}
  1. Multiply Conjugate: Multiply the numerator and the denominator by the conjugate of the denominator.\newline(9×(5−5))/((5+5)×(5−5))(9 \times (5 - \sqrt{5})) / ((5 + \sqrt{5}) \times (5 - \sqrt{5}))
  2. Simplify Numerator: Simplify the numerator by distributing 99 to both terms in the conjugate.9×5−9×59 \times 5 - 9 \times \sqrt{5}=45−9×5= 45 - 9 \times \sqrt{5}
  3. Simplify Denominator: Simplify the denominator by using the difference of squares formula, which states that (a+b)(a−b)=a2−b2(a + b)(a - b) = a^2 - b^2.(5)2−(5)2(5)^2 - (\sqrt{5})^2=25−5= 25 - 5=20= 20
  4. Combine Terms: Combine the simplified numerator and denominator. (45−9×5)/20(45 - 9 \times \sqrt{5}) / 20
  5. Split Fraction: The expression is now rationalized and simplified. We can split the fraction into two parts to make it clearer. 4520−9⋅520\frac{45}{20} - \frac{9 \cdot \sqrt{5}}{20}
  6. Reduce Fractions: Simplify each term by reducing the fractions.\newline4520=94\frac{45}{20} = \frac{9}{4}\newline9⋅520=920⋅5\frac{9 \cdot \sqrt{5}}{20} = \frac{9}{20} \cdot \sqrt{5}
  7. Final Expression: Write the final simplified and rationalized expression. 94−(920)⋅5\frac{9}{4} - \left(\frac{9}{20}\right) \cdot \sqrt{5}

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