Evaluate \dfrac { 15 }{ \sqrt { 10 } +\sqrt { 20 } +\sqrt { 40 } -\sqrt { 5 } -\sqrt { 80 }  }

Foundation

Mathematics Foundation

Surds

Properties of Real Number

Rationalisation

Question

Evaluate

\dfrac { 15 }{ \sqrt { 10 } +\sqrt { 20 } +\sqrt { 40 } -\sqrt { 5 } -\sqrt { 80 }&nbsp; }

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\frac{15}{\sqrt{10} + \sqrt{20} + \sqrt{40} - \sqrt{5} - \sqrt{80},}

= \frac{15}{\sqrt{10} + 2 \sqrt{5} + 2 \sqrt{10} - \sqrt{5} - 4 \sqrt{5}}

= \frac{15}{3 \sqrt{10} - 3 \sqrt{5}}

= \frac{5}{\sqrt{10} - \sqrt{5}}

= \frac{\sqrt{5}}{(\sqrt{2} - 1)}

= \frac{\sqrt{5} (\sqrt{2} + 1)}{(\sqrt{2} - 1) (\sqrt{2} + 1)}

= \sqrt{10} + \sqrt{5}