Educational

Simplify Surd Find the positive square root \[ \sqrt{7 + \sqrt{48}} \] (a) \(7 + 2\sqrt{3}\) \quad (b) \(7 + \sqrt{3}\) \quad (c) \(2 + \sqrt{3}\) \quad (d) \(3 + \sqrt{2}\) Solution: \[ \sqrt{48} = 4\sqrt{3} \] \[ \sqrt{7 + 4\sqrt{3}} = \sqrt{a} + \sqrt{b} \] \[ a + b = 7,\quad ab = 12

Find the Value Find the value upto three decimal places If \[ \sqrt{2} = 1.414 \] Find: \[ \sqrt{6} – \sqrt{3} \] (a) 0.235 \quad (b) 0.717 \quad (c) 1.414 \quad (d) 0.471 Solution: \[ \sqrt{6} = \sqrt{2 \times 3} = \sqrt{2}\sqrt{3} \] \[ \sqrt{3} \approx 1.732 \] \[ \sqrt{6} = 1.414 \times 1.732 =

Find the Value Find the value If \[ \sqrt{2} = 1.4142,\ \text{then} \quad \sqrt{\frac{\sqrt{2} – 1}{\sqrt{2} + 1}} \] (a) 0.1718 \quad (b) 5.8282 \quad (c) 0.4142 \quad (d) 2.4142 Solution: \[ \frac{\sqrt{2} – 1}{\sqrt{2} + 1} \times \frac{\sqrt{2} – 1}{\sqrt{2} – 1} = \frac{(\sqrt{2} – 1)^2}{2 – 1} \] \[ = (\sqrt{2} – 1)^2

Simplify Surd Find the value \[ \sqrt{5 + 2\sqrt{6}} \] (a) \(\sqrt{3} – \sqrt{2}\) \quad (b) \(\sqrt{3} + \sqrt{2}\) \quad (c) \(\sqrt{5} + \sqrt{6}\) \quad (d) none of these Solution: \[ \sqrt{5 + 2\sqrt{6}} = \sqrt{a} + \sqrt{b} \] \[ (\sqrt{a} + \sqrt{b})^2 = a + b + 2\sqrt{ab} \] \[ a + b =

Simplify Surd Find the value \[ \sqrt{3 – 2\sqrt{2}} \] (a) \(\sqrt{2} – 1\) \quad (b) \(\sqrt{2} + 1\) \quad (c) \(\sqrt{3} – \sqrt{2}\) \quad (d) \(\sqrt{3} + \sqrt{2}\) Solution: \[ \sqrt{3 – 2\sqrt{2}} = \sqrt{a} – \sqrt{b} \] \[ (\sqrt{a} – \sqrt{b})^2 = a + b – 2\sqrt{ab} \] \[ a + b =

Find the Value Find the value If \[ x = \sqrt[3]{2 + \sqrt{3}} \] (a) 2 \quad (b) 4 \quad (c) 8 \quad (d) 9 Solution: \[ x^3 = 2 + \sqrt{3} \] \[ \frac{1}{x^3} = \frac{1}{2 + \sqrt{3}} \times \frac{2 – \sqrt{3}}{2 – \sqrt{3}} = 2 – \sqrt{3} \] \[ x^3 + \frac{1}{x^3} =

Find x and y Find the values of \(x\) and \(y\) If \[ \frac{5 – \sqrt{3}}{2 + \sqrt{3}} = x + y\sqrt{3} \] (a) \(x = 13, y = -7\) \quad (b) \(x = -13, y = 7\) \quad (c) \(x = -13, y = -7\) \quad (d) \(x = 13, y = 7\) Solution:

Simplify Expression Simplify If \[ \frac{1}{\sqrt{9} – \sqrt{8}} \text{ is equal to } \] (a) \(3 + 2\sqrt{2}\) \quad (b) \(\frac{1}{3 + 2\sqrt{2}}\) \quad (c) \(3 – 2\sqrt{2}\) \quad (d) \(\frac{3}{2} – \sqrt{2}\) Solution: \[ = \frac{1}{3 – 2\sqrt{2}} \times \frac{3 + 2\sqrt{2}}{3 + 2\sqrt{2}} \] \[ = \frac{3 + 2\sqrt{2}}{9 – 8} = 3

Find the Value Find the value If \[ x = \frac{\sqrt{3} – \sqrt{2}}{\sqrt{3} + \sqrt{2}}, \quad y = \frac{\sqrt{3} + \sqrt{2}}{\sqrt{3} – \sqrt{2}}, \] then \[ x^2 + xy + y^2 = \ ? \] Solution: \[ x = \frac{(\sqrt{3} – \sqrt{2})^2}{3 – 2} = 5 – 2\sqrt{6} \] \[ y = \frac{(\sqrt{3} + \sqrt{2})^2}{3

Find the Value Find the value If \[ x = \frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} – \sqrt{3}} \quad \text{and} \quad y = \frac{\sqrt{5} – \sqrt{3}}{\sqrt{5} + \sqrt{3}}, \] then \[ x + y + xy = \ ? \] Solution: \[ x = \frac{(\sqrt{5} + \sqrt{3})^2}{5 – 3} = \frac{5 + 3 + 2\sqrt{15}}{2} = 4 +