Ejercicio 238 – Solucionario Baldor

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DESCARGA

CAPITULO XXXI

R a d i c a l e s
Suma y Resta de radicales

Ejercicio 238

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$\begin{matrix} \sqrt{45} - \sqrt{27} - \sqrt{20} & = \sqrt{3^{2} .5} - \sqrt{3^{2} .3} - \sqrt{2^{2} .5} \\ = 3 \sqrt{5} - 3 \sqrt{3} - 2 \sqrt{5} \\ = \sqrt{5} (3 - 2) - 3 \sqrt{3} \\ = \sqrt{5} - 3 \sqrt{3} \end{matrix}$
$\begin{matrix} \sqrt{175} + \sqrt{243} - \sqrt{63} - 2 \sqrt{75} & = \sqrt{5^{2} .7} + \sqrt{9^{2} .3} - \sqrt{3^{2} .7} - 2 \sqrt{5^{2} .3} \\ = 5 \sqrt{7} + 9 \sqrt{3} - 3 \sqrt{7} - 10 \sqrt{3} \\ = \sqrt{7} (5 - 3) + \sqrt{3} (9 - 10) \\ = 2 \sqrt{7} - \sqrt{3} \end{matrix}$
$\begin{matrix} \sqrt{80} - 2 \sqrt{252} + 3 \sqrt{405} - 3 \sqrt{500} & = \sqrt{4^{2} .5} - 2 \sqrt{2^{2} . 3^{2} .7} + 3 \sqrt{9^{2} .5} - 3 \sqrt{5.1 0^{2}} \\ = 4 \sqrt{5} - 2.2.3 \sqrt{7} + 3.9 \sqrt{5} - 3.10 \sqrt{5} \\ = 4 \sqrt{5} - 12 \sqrt{7} + 27 \sqrt{5} - 30 \sqrt{5} \\ = \sqrt{5} (4 + 27 - 30) - 12 \sqrt{7} \\ = \sqrt{5} - 12 \sqrt{7} \end{matrix}$
$\begin{matrix} 7 \sqrt{450} - 4 \sqrt{320} + 3 \sqrt{80} - 5 \sqrt{800} & = 7 \sqrt{3^{2} . 5^{2} .2} - 4 \sqrt{2^{6} .5} + 3 \sqrt{4^{2} .5} - 5 \sqrt{2^{4} .2. 5^{2}} \\ = 7.3.5 \sqrt{2} - 4. 2^{3} \sqrt{5} + 3.4 \sqrt{5} - 5. 2^{2} .5 \sqrt{2} \\ = 105 \sqrt{2} - 32 \sqrt{5} + 12 \sqrt{5} - 100 \sqrt{2} \\ = \sqrt{2} (105 - 100) + \sqrt{5} (12 - 32) \\ = 5 \sqrt{2} - 20 \sqrt{5} \end{matrix}$
$\begin{matrix} \frac{1}{2} \sqrt{12} - \frac{1}{3} \sqrt{18} + \frac{3}{4} \sqrt{48} + \frac{1}{6} \sqrt{72} & = \frac{1}{2} \sqrt{2^{2} .3} - \frac{1}{3} \sqrt{3^{2} .2} + \frac{3}{4} \sqrt{4^{2} .3} + \frac{1}{6} \sqrt{2^{2} .2. 3^{2}} \\ = \frac{1}{2} . 2 \sqrt{3} - \frac{1}{3} . 3 \sqrt{2} + \frac{3}{4} . 4 \sqrt{3} + \frac{1}{6} . 2.3 \sqrt{2} \\ = \sqrt{3} - \sqrt{2} + 3 \sqrt{3} + \sqrt{2} \\ = \sqrt{3} (1 + 3) + \sqrt{2} (1 - 1) \\ = 4 \sqrt{3} \end{matrix}$
$\begin{matrix} \frac{3}{4} \sqrt{176} - \frac{2}{3} \sqrt{45} + \frac{1}{8} \sqrt{320} + \frac{1}{5} \sqrt{275} & = \frac{3}{4} \sqrt{4^{2} .11} - \frac{2}{3} \sqrt{3^{2} .5} + \frac{1}{8} \sqrt{2^{6} .5} + \frac{1}{5} \sqrt{5^{2} .11} \\ = \frac{3}{4} . 4 \sqrt{11} - \frac{2}{3} . 3 \sqrt{5} + \frac{1}{8} . 2^{3} \sqrt{5} + \frac{1}{5} . 5 \sqrt{11} \\ = 3 \sqrt{11} - 2 \sqrt{5} + \sqrt{5} + \sqrt{11} \\ = \sqrt{11} (3 + 1) + \sqrt{5} (1 - 2) \\ = 4 \sqrt{11} - \sqrt{5} \end{matrix}$
$\begin{matrix} \frac{1}{7} \sqrt{147} - \frac{1}{5} \sqrt{700} + \frac{1}{10} \sqrt{28} + \frac{1}{3} \sqrt{2187} & = \frac{1}{7} \sqrt{3. 7^{2}} - \frac{1}{5} \sqrt{2^{2} . 5^{2} .7} + \frac{1}{10} \sqrt{2^{2} .7} + \frac{1}{3} \sqrt{3^{6} .3} \\ = \frac{1}{7} . 7 \sqrt{3} - \frac{1}{7} .2. 5 \sqrt{7} + \frac{1}{} . 2 \sqrt{7} + \frac{1}{3} . 3^{} \sqrt{3} \\ = \sqrt{3} - 2 \sqrt{7} + \frac{1}{5} \sqrt{7} + 9 \sqrt{3} \\ = \sqrt{3} (1 + 9) + \sqrt{7} (\frac{1}{5} - 2) \\ = 10 \sqrt{3} - \frac{9}{5} \sqrt{7} \end{matrix}$
$\begin{matrix} \sqrt{\frac{1}{3}} - \sqrt{\frac{1}{2}} + \sqrt{\frac{3}{4}} & = \sqrt{\frac{1.3}{3.3}} - \sqrt{\frac{1.2}{2.2}} + \sqrt{\frac{3}{2^{2}}} \\ = \sqrt{\frac{3}{3^{2}}} - \sqrt{\frac{2}{2^{2}}} + \sqrt{\frac{3}{2^{2}}} \\ = \frac{\sqrt{3}}{3} - \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \\ = \sqrt{3} (\frac{1}{3} + \frac{1}{2}) - \frac{\sqrt{2}}{2} \\ = \sqrt{3} (\frac{2 + 3}{6}) - \frac{\sqrt{2}}{2} \\ = \frac{5}{6} \sqrt{3} - \frac{\sqrt{2}}{2} \end{matrix}$
$\begin{matrix} \sqrt{\frac{9}{5}} - \sqrt{\frac{1}{6}} - \sqrt{\frac{1}{20}} + \sqrt{6} & = \sqrt{\frac{9.5}{5.5}} - \sqrt{\frac{1.6}{6.6}} - \sqrt{\frac{1.5}{2^{2} .5.5}} + \sqrt{6} \\ = \sqrt{\frac{3^{2} .5}{5^{2}}} - \sqrt{\frac{1.6}{6^{2}}} - \sqrt{\frac{5}{2^{2} . 5^{2}}} + \sqrt{6} \\ = \frac{3 \sqrt{5}}{5} - \frac{\sqrt{6}}{6} - \frac{\sqrt{5}}{10} + \sqrt{6} \\ = \frac{\sqrt{5}}{5} (3 - \frac{1}{2}) + \sqrt{6} (1 - \frac{1}{6}) \\ = \frac{\sqrt{5}}{5} (\frac{6 - 1}{2}) + \sqrt{6} (\frac{6 - 1}{6}) \\ = \frac{\sqrt{5}}{5} (\frac{5}{2}) + \sqrt{6} (\frac{5}{6}) \\ = \frac{\sqrt{5}}{2} - \frac{5 \sqrt{6}}{6} \end{matrix}$
$\begin{matrix} \frac{5}{3} \sqrt{\frac{3}{5}} - \frac{1}{2} \sqrt{\frac{3}{4}} - 5 \sqrt{\frac{1}{15}} + 3 \sqrt{\frac{1}{12}} & = \frac{5}{3} \sqrt{\frac{3.5}{5.5}} - \frac{1}{2} \sqrt{\frac{3}{2^{2}}} - 5 \sqrt{\frac{1.15}{15.15}} + 3 \sqrt{\frac{1.3}{3. 2^{2} .3}} \\ = \frac{5}{3} \sqrt{\frac{15}{5^{2}}} - \frac{1}{2} \sqrt{\frac{3}{2^{2}}} - 5 \sqrt{\frac{15}{1 5^{2}}} + 3 \sqrt{\frac{3}{3^{2} . 2^{2}}} \\ = \frac{5}{3. 5} \sqrt{15} - \frac{1}{2.2} \sqrt{3} - \frac{5}{} \sqrt{15} + \frac{3}{3 .2} \sqrt{3} \\ = \frac{1}{3} \sqrt{15} - \frac{1}{4} \sqrt{3} - \frac{1}{3} \sqrt{15} + \frac{1}{2} \sqrt{3} \\ = \sqrt{15} (\frac{1}{3} - \frac{1}{3}) + \sqrt{3} (\frac{1}{2} - \frac{1}{4}) \\ = \sqrt{3} (\frac{2 - 1}{4}) \\ = \frac{1}{4} \sqrt{3} \end{matrix}$
$\begin{matrix} 5 \sqrt{128} - \frac{1}{3} \sqrt{\frac{1}{3}} - 5 \sqrt{98} + \sqrt{\frac{1}{27}} & = 5 \sqrt{2^{6} .2} - \frac{1}{3} \sqrt{\frac{1.3}{3.3}} - 5 \sqrt{7^{2} .2} + \sqrt{\frac{1.3}{3^{3} .3}} \\ = 5. 2^{3} \sqrt{2} - \frac{1}{3} \sqrt{\frac{3}{3^{2}}} - 5.7 \sqrt{2} + \sqrt{\frac{3}{3^{4}}} \\ = 40 \sqrt{2} - \frac{1}{3.3} \sqrt{3} - 35 \sqrt{2} + \frac{1}{3^{2}} \sqrt{3} \\ = \sqrt{2} (40 - 35) \\ = 5 \sqrt{2} \end{matrix}$
$\begin{matrix} 2 \sqrt{700} - 15 \sqrt{\frac{1}{45}} + 4 \sqrt{\frac{5}{16}} - 56 \sqrt{\frac{1}{7}} & = 2 \sqrt{7.1 0^{2}} - 15 \sqrt{\frac{1.5}{3^{2} .5.5}} + 4 \sqrt{\frac{5}{4^{2}}} - 56 \sqrt{\frac{1.7}{7.7}} \\ = 2.10 \sqrt{7} - 15 \sqrt{\frac{5}{3^{2} . 5^{2}}} + 4 . \frac{1}{4} \sqrt{5} - 56 \sqrt{\frac{7}{7^{2}}} \\ = 20 \sqrt{7} - 15 . \frac{1}{3.5} \sqrt{5} + \sqrt{5} - . \frac{1}{7} \sqrt{7} \\ = 20 \sqrt{7} - \sqrt{5} + \sqrt{5} - 8 \sqrt{7} \\ = \sqrt{7} (20 - 8) \\ = 12 \sqrt{7} \end{matrix}$
$\begin{matrix} \sqrt{25 a x^{2}} + \sqrt{49 b} - \sqrt{9 a x^{2}} & = \sqrt{5^{2} a x^{2}} + \sqrt{7^{2} b} - \sqrt{3^{2} a x^{2}} \\ = 5 x \sqrt{a} + 7 \sqrt{b} - 3 x \sqrt{a} \\ = x \sqrt{a} (5 - 3) + 7 \sqrt{b} \\ = 2 x \sqrt{a} + 7 \sqrt{b} \end{matrix}$
$\begin{matrix} 2 \sqrt{m^{2} n} - \sqrt{9 m^{2} n} + \sqrt{16 m n^{2}} - \sqrt{4 m n^{2}} & = 2 m \sqrt{n} - \sqrt{3^{2} m^{2} n} + \sqrt{4^{2} m n^{2}} - \sqrt{2^{2} m n^{2}} \\ = 2 m \sqrt{n} - 3 m \sqrt{n} + 4 n \sqrt{m} - 2 n \sqrt{m} \\ = 2 n \sqrt{m} - m \sqrt{n} \end{matrix}$
$\begin{matrix} a \sqrt{320 x} - 7 \sqrt{5 a^{2} x} - (a - 4 b) \sqrt{5 x} & = a \sqrt{2^{6} .5 x} - 7 a \sqrt{5 x} - (a - 4 b) \sqrt{5 x} \\ = a \sqrt{2^{6} .5 x} - 7 a \sqrt{5 x} - (a - 4 b) \sqrt{5 x} \\ = a . 2^{3} \sqrt{5 x} - 7 a \sqrt{5 x} - (a - 4 b) \sqrt{5 x} \\ = 8 a \sqrt{5 x} - 7 a \sqrt{5 x} - (a - 4 b) \sqrt{5 x} \\ = \sqrt{5 x} [8 a - 7 a - (a - 4 b)] \\ = \sqrt{5 x} [8 a - 7 a - a + 4 b] \\ = 4 b \sqrt{5 x} \end{matrix}$
$\begin{matrix} \sqrt{9 x - 9} + \sqrt{4 x - 4} - 5 \sqrt{x - 1} & = \sqrt{9 (x - 1)} + \sqrt{4 (x - 1)} - 5 \sqrt{x - 1} \\ = \sqrt{3^{2} (x - 1)} + \sqrt{2^{2} (x - 1)} - 5 \sqrt{x - 1} \\ = 3 \sqrt{x - 1} + 2 \sqrt{x - 1} - 5 \sqrt{x - 1} \\ = 0 \end{matrix}$
$\begin{matrix} 2 \sqrt{a^{4} x + 3 a^{4} y} - a^{2} \sqrt{9 x + 27 y} + \sqrt{25 a^{4} x + 75 a^{4} y} & = 2 \sqrt{a^{4} (x + 3 y)} - a^{2} \sqrt{9 (x + 3 y)} + \sqrt{25 a^{4} (x + 3 y)} \\ = 2 a^{2} \sqrt{x + 3 y} - a^{2} \sqrt{3^{2} (x + 3 y)} + \sqrt{5^{2} a^{4} (x + 3 y)} \\ = 2 a^{2} \sqrt{x + 3 y} - 3 a^{2} \sqrt{x + 3 y} + 5 a^{2} \sqrt{x + 3 y} \\ = 4 a^{2} \sqrt{x + 3 y} \end{matrix}$
$\begin{matrix} 3 a \sqrt{\frac{a + 1}{a^{2}}} - \sqrt{4 a + 4} + (a + 1) \sqrt{\frac{1}{a + 1}} & = 3 a . \frac{1}{a} \sqrt{a + 1} - \sqrt{4 (a + 1)} + (a + 1) \sqrt{\frac{1}{a + 1} . \frac{a + 1}{a + 1}} \\ = 3 \sqrt{a + 1} - \sqrt{2^{2} (a + 1)} + (a + 1) \sqrt{\frac{a + 1}{{(a + 1)}^{2}}} \\ = 3 \sqrt{a + 1} - 2 \sqrt{a + 1} + (a + 1) . \frac{1}{a + 1} \sqrt{a + 1} \\ = 3 \sqrt{a + 1} - 2 \sqrt{a + 1} + \sqrt{a + 1} \\ = 2 \sqrt{a + 1} \end{matrix}$
$\begin{matrix} (a - b) \sqrt{\frac{a + b}{a - b}} - (a + b) \sqrt{\frac{a - b}{a + b}} + (2 a - 2 b) \sqrt{\frac{1}{a - b}} & = (a - b) \sqrt{\frac{a + b}{a - b} . \frac{a - b}{a - b}} - (a + b) \sqrt{\frac{a - b}{a + b} . \frac{a + b}{a + b}} + 2 (a - b) \sqrt{\frac{1}{a - b} . \frac{a - b}{a - b}} \\ = (a - b) \sqrt{\frac{a^{2} - b^{2}}{{(a - b)}^{2}}} - (a + b) \sqrt{\frac{a^{2} - b^{2}}{{(a + b)}^{2}}} + 2 (a - b) \sqrt{\frac{a - b}{{(a - b)}^{2}}} \\ = (a - b) . \frac{1}{a - b} \sqrt{a^{2} - b^{2}} - (a + b) . \frac{1}{a + b} \sqrt{a^{2} - b^{2}} + 2 (a - b) . \frac{1}{a - b} \sqrt{a - b} \\ = \sqrt{a^{2} - b^{2}} - \sqrt{a^{2} - b^{2}} + 2 \sqrt{a - b} \\ = 2 \sqrt{a - b} \end{matrix}$