Educational

Determine a and b Determine rational numbers \(a\) and \(b\) \[ \frac{4 + \sqrt{2}}{2 + \sqrt{2}} = a – \sqrt{b} \] Solution: \[ \frac{4 + \sqrt{2}}{2 + \sqrt{2}} \times \frac{2 – \sqrt{2}}{2 – \sqrt{2}} \] \[ = \frac{(4 + \sqrt{2})(2 – \sqrt{2})}{(2)^2 – (\sqrt{2})^2} \] \[ = \frac{8 – 4\sqrt{2} + 2\sqrt{2} – 2}{4 – […]

Determine a and b Determine rational numbers \(a\) and \(b\) \[ \frac{\sqrt{3} – 1}{\sqrt{3} + 1} = a – b\sqrt{3} \] Solution: \[ \frac{\sqrt{3} – 1}{\sqrt{3} + 1} \times \frac{\sqrt{3} – 1}{\sqrt{3} – 1} \] \[ = \frac{(\sqrt{3} – 1)^2}{(\sqrt{3})^2 – (1)^2} \] \[ = \frac{3 – 2\sqrt{3} + 1}{3 – 1} \] \[ =

Simplify Surds Expression 🎥 Video Solution: 📘 Simplify: \[ \frac{2}{\sqrt{5}+\sqrt{3}} + \frac{1}{\sqrt{3}+\sqrt{2}} + \frac{3}{\sqrt{5}+\sqrt{2}} \] ✏️ Solution (Rationalisation Flow): \[ = \frac{2(\sqrt{5}-\sqrt{3})}{(\sqrt{5}+\sqrt{3})(\sqrt{5}-\sqrt{3})} + \frac{\sqrt{3}-\sqrt{2}}{(\sqrt{3}+\sqrt{2})(\sqrt{3}-\sqrt{2})} + \frac{3(\sqrt{5}-\sqrt{2})}{(\sqrt{5}+\sqrt{2})(\sqrt{5}-\sqrt{2})} \] \[ = \frac{2(\sqrt{5}-\sqrt{3})}{5-3} + \frac{\sqrt{3}-\sqrt{2}}{3-2} + \frac{3(\sqrt{5}-\sqrt{2})}{5-2} \] \[ = (\sqrt{5}-\sqrt{3}) + (\sqrt{3}-\sqrt{2}) + (\sqrt{5}-\sqrt{2}) \] \[ = 2\sqrt{5} – 2\sqrt{2} \] ✅ Final Answer: \(2(\sqrt{5} – \sqrt{2})\)

Simplify Surds Expression 🎥 Video Solution: 📘 Simplify: \[ \frac{1}{2+\sqrt{3}} + \frac{2}{\sqrt{5}-\sqrt{3}} + \frac{1}{2-\sqrt{5}} \] ✏️ Solution (Rationalisation Flow): \[ = \frac{2-\sqrt{3}}{(2+\sqrt{3})(2-\sqrt{3})} + \frac{2(\sqrt{5}+\sqrt{3})}{(\sqrt{5}-\sqrt{3})(\sqrt{5}+\sqrt{3})} + \frac{2+\sqrt{5}}{(2-\sqrt{5})(2+\sqrt{5})} \] \[ = \frac{2-\sqrt{3}}{4-3} + \frac{2(\sqrt{5}+\sqrt{3})}{5-3} + \frac{2+\sqrt{5}}{4-5} \] \[ = (2-\sqrt{3}) + (\sqrt{5}+\sqrt{3}) – (2+\sqrt{5}) \] \[ = 0 \] ✅ Final Answer: \(0\) Next Question / Full

Simplify Surds Expression 🎥 Video Solution: 📘 Simplify: \[ \frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}} + \frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}} \] ✏️ Solution: \[ = \frac{(\sqrt{5}+\sqrt{3})^2 + (\sqrt{5}-\sqrt{3})^2}{5 – 3} \] \[ = \frac{(5+3+2\sqrt{15}) + (5+3-2\sqrt{15})}{2} \] \[ = \frac{16}{2} \] \[ = 8 \] ✅ Final Answer: \(8\) Next Question / Full Exercise

Rationalise (3√5 + √3)/(√5 − √3) 🎥 Video Solution: 📘 Rationalise & Simplify: \[ \frac{3\sqrt{5} + \sqrt{3}}{\sqrt{5} – \sqrt{3}} \] ✏️ Solution: \[ = \frac{3\sqrt{5} + \sqrt{3}}{\sqrt{5} – \sqrt{3}} \times \frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} + \sqrt{3}} \] \[ = \frac{(3\sqrt{5} + \sqrt{3})(\sqrt{5} + \sqrt{3})}{5 – 3} \] \[ = \frac{15 + 3\sqrt{15} + \sqrt{15} + 3}{2}

Rationalise (√3 − √2)/(√3 + √2) 🎥 Video Solution: 📘 Rationalise & Simplify: \[ \frac{\sqrt{3} – \sqrt{2}}{\sqrt{3} + \sqrt{2}} \] ✏️ Solution: \[ = \frac{\sqrt{3} – \sqrt{2}}{\sqrt{3} + \sqrt{2}} \times \frac{\sqrt{3} – \sqrt{2}}{\sqrt{3} – \sqrt{2}} \] \[ = \frac{(\sqrt{3} – \sqrt{2})^2}{3 – 2} \] \[ = 3 + 2 – 2\sqrt{6} \] \[ = 5

Rationalise (4√3 + 5√2)/(√48 + √18) 🎥 Video Solution: 📘 Rationalise & Simplify: \[ \frac{4\sqrt{3} + 5\sqrt{2}}{\sqrt{48} + \sqrt{18}} \] ✏️ Solution: \[ \sqrt{48} = 4\sqrt{3}, \quad \sqrt{18} = 3\sqrt{2} \] \[ = \frac{4\sqrt{3} + 5\sqrt{2}}{4\sqrt{3} + 3\sqrt{2}} \] \[ \times \frac{4\sqrt{3} – 3\sqrt{2}}{4\sqrt{3} – 3\sqrt{2}} \] \[ = \frac{(4\sqrt{3} + 5\sqrt{2})(4\sqrt{3} – 3\sqrt{2})}{(4\sqrt{3})^2 –

Rationalise (1 + √2)/(3 − 2√2) 🎥 Video Solution: 📘 Rationalise & Simplify: \[ \frac{1 + \sqrt{2}}{3 – 2\sqrt{2}} \] ✏️ Solution: \[ = \frac{1 + \sqrt{2}}{3 – 2\sqrt{2}} \times \frac{3 + 2\sqrt{2}}{3 + 2\sqrt{2}} \] \[ = \frac{(1 + \sqrt{2})(3 + 2\sqrt{2})}{3^2 – (2\sqrt{2})^2} \] \[ = \frac{3 + 2\sqrt{2} + 3\sqrt{2} + 4}{9

Rationalise (5 + 2√3)/(7 + 4√3) 🎥 Video Solution: 📘 Rationalise & Simplify: \[ \frac{5 + 2\sqrt{3}}{7 + 4\sqrt{3}} \] ✏️ Solution: \[ = \frac{5 + 2\sqrt{3}}{7 + 4\sqrt{3}} \times \frac{7 – 4\sqrt{3}}{7 – 4\sqrt{3}} \] \[ = \frac{(5 + 2\sqrt{3})(7 – 4\sqrt{3})}{7^2 – (4\sqrt{3})^2} \] \[ = \frac{35 – 20\sqrt{3} + 14\sqrt{3} – 24}{49