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Simplify. Rationalize the denominator.\newline84−5\frac{8}{4 - \sqrt{5}}

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Full solution

Q. Simplify. Rationalize the denominator.\newline84−5\frac{8}{4 - \sqrt{5}}
  1. Identify Conjugate of Denominator: Identify the conjugate of the denominator.\newlineThe conjugate of a−ba - \sqrt{b} is a+ba + \sqrt{b}. Therefore, the conjugate of 4−54 - \sqrt{5} is 4+54 + \sqrt{5}.
  2. Multiply by Conjugate: Multiply the numerator and the denominator by the conjugate of the denominator.\newlineTo rationalize the denominator, we multiply the expression by (4+5)/(4+5)(4 + \sqrt{5})/(4 + \sqrt{5}).\newline(8/(4−5))⋅((4+5)/(4+5))(8/(4 - \sqrt{5})) \cdot ((4 + \sqrt{5})/(4 + \sqrt{5}))
  3. Apply Distributive Property: Apply the distributive property to multiply the numerators.\newline8×(4+5)=8×4+8×5=32+8×58 \times (4 + \sqrt{5}) = 8\times4 + 8\times\sqrt{5} = 32 + 8\times\sqrt{5}
  4. Apply Difference of Squares: Apply the difference of squares formula to multiply the denominators.\newline(4−5)×(4+5)=42−(5)2=16−5=11(4 - \sqrt{5}) \times (4 + \sqrt{5}) = 4^2 - (\sqrt{5})^2 = 16 - 5 = 11
  5. Write Simplified Expression: Write the simplified expression with a rationalized denominator.\newline(32+8511)(\frac{32 + 8\sqrt{5}}{11})

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