Question

1. ( 3sqrt{\frac{5}{7}} - 6sqrt{\frac{7}{5}} ) 23. ( 4sqrt{24}(2sqrt{6} - 3sqrt{2}) ) 24. find the equation of the line that is perpendicular to ( 3y + x = -2 ) and passes through the point ((-2, -5)).

Explanation:

Step 22.1: Rationalize each radical term

$3\sqrt{\frac{5}{7}} = 3\cdot\frac{\sqrt{35}}{7} = \frac{3\sqrt{35}}{7}$, $6\sqrt{\frac{7}{5}} = 6\cdot\frac{\sqrt{35}}{5} = \frac{6\sqrt{35}}{5}$

Step 22.2: Find common denominator and subtract

Common denominator is 35: $\frac{15\sqrt{35}}{35} - \frac{42\sqrt{35}}{35}$

Step 23.1: Simplify $\sqrt{24}$

$\sqrt{24} = 2\sqrt{6}$, so $4\sqrt{24} = 8\sqrt{6}$

Step 23.2: Distribute the term

$8\sqrt{6}(2\sqrt{6} - 3\sqrt{2}) = 8\sqrt{6}\cdot2\sqrt{6} - 8\sqrt{6}\cdot3\sqrt{2}$

Step 23.3: Calculate each product

$16\cdot6 - 24\sqrt{12} = 96 - 24\cdot2\sqrt{3} = 96 - 48\sqrt{3}$

Step 24.1: Find slope of given line

Rewrite $3y + x = -2$ as $y = -\frac{1}{3}x - \frac{2}{3}$, slope $m_1 = -\frac{1}{3}$

Step 24.2: Find perpendicular slope

Perpendicular slope $m_2 = 3$ (negative reciprocal)

Step 24.3: Use point-slope form

$y - (-5) = 3(x - (-2))$

Step 24.4: Simplify to slope-intercept form

$y + 5 = 3x + 6 \implies y = 3x + 1$

Answer:

1. $\boldsymbol{-\frac{27\sqrt{35}}{35}}$
2. $\boldsymbol{96 - 48\sqrt{3}}$
3. $\boldsymbol{y = 3x + 1}$ (or $3x - y + 1 = 0$)