(m.) $\frac{4-2 \sqrt{10}}{\sqrt{50}}$

Full solution

Q. (m.)

\frac{4-2 \sqrt{10}}{\sqrt{50}}

Apply Quotient Rule: Apply the quotient rule of radicals to simplify $\sqrt{50}$ . $\sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2} = 5\sqrt{2}$
Rewrite with Simplified Denominator: Rewrite the original expression with the simplified denominator. $\newline$ $(4 - 2\sqrt{10}) / (\sqrt{50}) = (4 - 2\sqrt{10}) / (5\sqrt{2})$
Rationalize Denominator: Rationalize the denominator by multiplying the numerator and the denominator by $\sqrt{2}$ . $\frac{4 - 2\sqrt{10}}{5\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{4\sqrt{2} - 2\sqrt{10} \times \sqrt{2}}{5\sqrt{2} \times \sqrt{2}}$
Simplify Numerator and Denominator: Simplify the numerator and the denominator. $(4\sqrt{2} - 2\sqrt{20}) / (5 \times 2) = (4\sqrt{2} - 2\sqrt{4 \times 5}) / 10$
Simplify Square Root: Simplify the square root in the numerator. $\sqrt{20} = \sqrt{4 \times 5} = \sqrt{4} \times \sqrt{5} = 2\sqrt{5}$
Substitute Simplified Square Root: Substitute the simplified square root back into the numerator. $\newline$ $(4\sqrt{2} - 2 \times 2\sqrt{5}) / 10 = (4\sqrt{2} - 4\sqrt{5}) / 10$
Final Simplification: Simplify the expression by dividing both terms in the numerator by $10$ . $\frac{4\sqrt{2}}{10}$ - $\frac{4\sqrt{5}}{10}$ = $\frac{2}{5}$ \sqrt{ $2$ } - $\frac{2}{5}$ \sqrt{ $5$ }