▶️ Ejercicio 228 álgebra - Solucionario Baldor ✅✅

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CAPITULO XXX

TEORIA DE LOS EXPONENTES

Ejercicio 228

Desarrollar:

$\begin{matrix} {(a^{\frac{1}{2}} + b^{\frac{1}{2}})}^{2} & = {(a^{\frac{1}{2}})}^{2} + 2 a^{\frac{1}{2}} b^{\frac{1}{2}} + {(b^{\frac{1}{2}})}^{2} \\ = a + 2 a^{\frac{1}{2}} b^{\frac{1}{2}} + b \end{matrix}$
$\begin{matrix} {(x^{\frac{3}{4}} - y^{\frac{1}{3}})}^{2} & = {(x^{\frac{3}{}})}^{2} - 2 x^{\frac{3}{4}} y^{\frac{1}{3}} + {(- y^{\frac{1}{3}})}^{2} \\ = x^{\frac{3}{2}} - 2 x^{\frac{3}{4}} y^{\frac{1}{3}} + y^{\frac{2}{3}} \end{matrix}$
$\begin{matrix} {(m^{- \frac{1}{2}} + 2 m)}^{2} & = {(m^{- \frac{1}{2}})}^{2} + 2.2 m^{- \frac{1}{2} + 1} + {(2 m)}^{2} \\ = m^{- 1} + 4 m^{\frac{1}{2}} + 4 m^{2} \end{matrix}$
$\begin{matrix} {(a^{- 2} b^{3} - a^{3} b^{- 2})}^{2} & = {(a^{- 2} b^{3})}^{2} - 2 a^{- 2 + 3} b^{3 - 2} + {(- a^{3} b^{- 2})}^{2} \\ = a^{- 4} b^{6} - 2 a b + a^{6} b^{- 4} \end{matrix}$
$\begin{matrix} {(a^{- 1} - 3 b^{- \frac{3}{4}})}^{2} & = {(a^{- 1})}^{2} - 2.3 a^{- 1} b^{- \frac{3}{4}} + {(- 3 b^{- \frac{3}{}})}^{2} \\ = a^{- 2} - 6 a^{- 1} b^{- \frac{3}{4}} + 9 b^{- \frac{3}{2}} \end{matrix}$
$\begin{matrix} {(a^{- 2} + \sqrt{b})}^{2} & = {(a^{- 2})}^{2} + 2 a^{- 2} \sqrt{b} + {(\sqrt{b})}^{2} \\ = a^{- 4} + 2 a^{- 2} \sqrt{b} + b \end{matrix}$
$\begin{matrix} {(\sqrt[4]{x^{3}} - y^{- \frac{1}{2}})}^{2} & = {(x^{3})}^{2} - 2 \sqrt[4]{x^{3}} y^{- \frac{1}{2}} + {(- y^{- \frac{1}{2}})}^{2} \\ = \sqrt{x^{3}} - 2 \sqrt[4]{x^{3}} y^{- \frac{1}{2}} + y^{- 1} \end{matrix}$
$\begin{matrix} {(m^{- 2} n^{\frac{1}{4}} - m^{\frac{1}{2}} n^{- 1})}^{2} & = {(m^{- 2} n^{\frac{1}{4}})}^{2} - m^{- 2 + \frac{1}{2}} n^{\frac{1}{4} - 1} + {(- m^{\frac{1}{2}} n^{- 1})}^{2} \\ = m^{- 4} n^{\frac{1}{2}} - m^{- \frac{3}{2}} n^{- \frac{3}{4}} + m n^{- 2} \end{matrix}$
$\begin{matrix} {(a^{\frac{1}{3}} + b^{\frac{1}{3}})}^{3} & = {(a^{\frac{1}{3}})}^{3} + 3 {(a^{\frac{1}{3}})}^{2} b^{\frac{1}{3}} + 3 a^{\frac{1}{3}} {(b^{\frac{1}{3}})}^{2} + {(b^{\frac{1}{3}})}^{3} \\ = a + 3 a^{\frac{2}{3}} b^{\frac{1}{3}} + 3 a^{\frac{1}{3}} b^{\frac{2}{3}} + b \end{matrix}$
$\begin{matrix} {(\sqrt[3]{x^{2}} - 3 y^{- 1})}^{3} & = {(x^{\frac{2}{3}})}^{3} - 3 {(x^{\frac{2}{3}})}^{2} (3 y^{- 1}) + 3 x^{\frac{2}{3}} {(3 y^{- 1})}^{2} - {(3 y^{- 1})}^{3} \\ = x^{2} - 9 x^{\frac{4}{3}} y^{- 1} + 3 x^{\frac{2}{3}} .9 y^{- 2} - 3^{3} y^{- 3} \\ = x^{2} - 9 x^{\frac{4}{3}} y^{- 1} + 27 x^{\frac{2}{3}} y^{- 2} - 27 y^{- 3} \end{matrix}$
$\begin{matrix} {(m^{\frac{2}{3}} + 4 n^{- \frac{3}{2}})}^{3} & = {(m^{\frac{2}{3}})}^{3} + 3 {(m^{\frac{2}{3}})}^{2} (4 n^{- \frac{3}{2}}) + 3 m^{\frac{2}{3}} {(4 n^{- \frac{3}{2}})}^{2} + {(4 n^{- \frac{3}{2}})}^{3} \\ = m^{2} + 12 m^{\frac{4}{3}} n^{- \frac{3}{2}} + 3 m^{\frac{2}{3}} 4^{2} n^{- 3} + 4^{3} n^{- \frac{9}{2}} \\ = m^{2} + 12 m^{\frac{4}{3}} n^{- \frac{3}{2}} + 48 m^{\frac{2}{3}} n^{- 3} + 64 n^{- \frac{9}{2}} \end{matrix}$
$\begin{matrix} {(2 a^{- 4} - 3 b^{- \frac{1}{2}})}^{3} & = {(2 a^{- 4})}^{3} - 3 {(2 a^{- 4})}^{2} (3 b^{- \frac{1}{2}}) + 3 (2 a^{- 4}) {(3 b^{- \frac{1}{2}})}^{2} - {(3 b^{- \frac{1}{2}})}^{3} \\ = 2^{3} a^{- 12} - 3^{2} . 2^{2} a^{- 8} b^{- \frac{1}{2}} + 3^{3} .2 a^{- 4} b^{- 1} - 3^{3} b^{- \frac{3}{2}} \\ = 8 a^{- 12} - 36 a^{- 8} b^{- \frac{1}{2}} + 54 a^{- 4} b^{- 1} - 27 b^{- \frac{3}{2}} \end{matrix}$
$\begin{matrix} {(\sqrt{x} - \sqrt[3]{y})}^{3} & = {(x^{\frac{1}{2}})}^{3} - 3 {(x^{\frac{1}{2}})}^{2} y^{\frac{1}{3}} + 3 x^{\frac{1}{2}} {(y^{\frac{1}{3}})}^{2} - {(y^{\frac{1}{3}})}^{3} \\ = x^{\frac{3}{2}} - 3 x y^{\frac{1}{3}} + 3 x^{\frac{1}{2}} y^{\frac{2}{3}} - y \end{matrix}$
$\begin{matrix} {(a^{\frac{1}{2}} + b^{\frac{2}{3}})}^{4} & = {(a^{\frac{1}{2}})}^{4} + 4 {(a^{\frac{1}{2}})}^{3} (b^{\frac{2}{3}}) + 6 {(a^{\frac{1}{2}})}^{2} {(b^{\frac{2}{3}})}^{2} + 4 (a^{\frac{1}{2}}) {(b^{\frac{2}{3}})}^{3} + {(b^{\frac{2}{3}})}^{4} \\ = a^{2} + 4 a^{\frac{3}{2}} b^{\frac{2}{3}} + 6 a b^{\frac{4}{3}} + 4 a^{\frac{1}{2}} b^{2} + b^{\frac{8}{3}} \end{matrix}$
$\begin{matrix} {(x^{- 2} - y^{- \frac{1}{3}})}^{4} & = {(x^{- 2})}^{4} - 4 {(x^{- 2})}^{3} (y^{- \frac{1}{3}}) + 6 {(x^{- 2})}^{2} {(y^{- \frac{1}{3}})}^{2} - 4 (x^{- 2}) {(y^{- \frac{1}{3}})}^{3} + {(y^{- \frac{1}{3}})}^{4} \\ = x^{- 8} - 4 x^{- 6} y^{- \frac{1}{3}} + 6 x^{- 4} y^{- \frac{2}{3}} - 4 x^{- 2} y^{- 1} + y^{- \frac{4}{3}} \end{matrix}$
$\begin{matrix} {(x^{\frac{1}{3}} + y^{- \frac{3}{4}})}^{5} & = {(x^{\frac{1}{3}})}^{5} + 5 {(x^{\frac{1}{3}})}^{4} (y^{- \frac{3}{4}}) + 10 {(x^{\frac{1}{3}})}^{3} {(y^{- \frac{3}{4}})}^{2} + 10 {(x^{\frac{1}{3}})}^{2} {(y^{- \frac{3}{4}})}^{3} + 5 (x^{\frac{1}{3}}) {(y^{- \frac{3}{4}})}^{4} + {(y^{- \frac{3}{4}})}^{5} \\ = x^{\frac{5}{3}} + 5 x^{\frac{4}{3}} y^{- \frac{3}{4}} + 10 x y^{- \frac{3}{2}} + 10 x^{\frac{2}{3}} y^{- \frac{9}{4}} + 5 x^{\frac{1}{3}} y^{- 3} + y^{- \frac{15}{4}} \end{matrix}$
$\begin{matrix} {(\sqrt{m} - \sqrt[3]{n})}^{5} & = {(m^{\frac{1}{2}})}^{5} - 5 {(m^{\frac{1}{2}})}^{4} (n^{\frac{1}{3}}) + 10 {(m^{\frac{1}{2}})}^{3} {(n^{\frac{1}{3}})}^{2} - 10 {(m^{\frac{1}{2}})}^{2} {(n^{\frac{1}{3}})}^{3} + 5 (m^{\frac{1}{2}}) {(n^{\frac{1}{3}})}^{4} - {(n^{\frac{1}{3}})}^{5} \\ = m^{\frac{5}{2}} - 5 m^{2} n^{\frac{1}{3}} + 10 m^{\frac{3}{2}} n^{\frac{2}{3}} - 10 m n + 5 m^{\frac{1}{2}} n^{\frac{4}{3}} - n^{\frac{5}{3}} \end{matrix}$
$\begin{matrix} {(a^{2} - 2 \sqrt{m})}^{6} & = {(a^{2})}^{6} - 6 {(a^{2})}^{5} (2 m^{\frac{1}{2}}) + 15 {(a^{2})}^{4} {(2 m^{\frac{1}{2}})}^{2} - 20 {(a^{2})}^{3} {(2 m^{\frac{1}{2}})}^{3} + 15 {(a^{2})}^{2} {(2 m^{\frac{1}{2}})}^{4} - 6 (a^{2}) {(2 m^{\frac{1}{2}})}^{5} + {(2 m^{\frac{1}{2}})}^{6} \\ = a^{12} - 12 a^{10} m^{\frac{1}{2}} + 15. 2^{2} a^{8} m - 20. 2^{3} a^{6} m^{\frac{3}{2}} + 15. 2^{4} a^{4} m^{2} - 6. 2^{5} a^{2} m^{\frac{5}{2}} + 2^{6} m^{3} \\ = a^{12} - 12 a^{10} m^{\frac{1}{2}} + 60 a^{8} m - 160 a^{6} m^{\frac{3}{2}} + 240 a^{4} m^{2} - 192 a^{2} m^{\frac{5}{2}} + 64 m^{3} \end{matrix}$
$\begin{matrix} {(x^{- 3} + \sqrt[4]{y})}^{5} & = {(x^{- 3})}^{5} + 5 {(x^{- 3})}^{4} (y^{\frac{1}{4}}) + 10 {(x^{- 3})}^{3} {(y^{\frac{1}{4}})}^{2} + 10 {(x^{- 3})}^{2} {(y^{\frac{1}{4}})}^{3} + 5 (x^{- 3}) {(y^{\frac{1}{4}})}^{4} + {(y^{\frac{1}{4}})}^{5} \\ = x^{- 15} + 5 x^{- 12} y^{\frac{1}{4}} + 10 x^{- 9} y^{\frac{1}{2}} + 10 x^{- 6} y^{\frac{3}{4}} + 5 x^{- 3} y + y^{\frac{5}{4}} \end{matrix}$
$\begin{matrix} {(a^{- 2} + 3 a^{- 1} + 2)}^{2} & = {(a^{- 2})}^{2} + {(3 a^{- 1})}^{2} + 2^{2} + 2 (a^{- 2}) (3 a^{- 1}) + 2 (a^{- 2}) (2) + 2 (3 a^{- 1}) (2) \\ = a^{- 4} + 9 a^{- 2} + 4 + 6 a^{- 3} + 4 a^{- 2} + 12 a^{- 1} \end{matrix}$
$\begin{matrix} {(x^{\frac{1}{2}} - x^{\frac{1}{4}} + 2 x^{- \frac{1}{4}})}^{2} & = {(x^{\frac{1}{2}})}^{2} + {(- x^{\frac{1}{4}})}^{2} + {(2 x^{- \frac{1}{4}})}^{2} + 2 (x^{\frac{1}{2}}) (- x^{\frac{1}{4}}) + 2 (x^{\frac{1}{2}}) (2 x^{- \frac{1}{4}}) + 2 (- x^{\frac{1}{4}}) (2 x^{- \frac{1}{4}}) \\ = x + x^{\frac{1}{2}} + 2^{2} x^{- \frac{1}{2}} - 2 x^{\frac{1}{2} + \frac{1}{4}} + 2^{2} x^{\frac{1}{2} - \frac{1}{4}} - 2^{2} x^{\frac{1}{4} - \frac{1}{4}} \\ = x + x^{\frac{1}{2}} + 4 x^{- \frac{1}{2}} - 2 x^{\frac{3}{4}} + 4 x^{\frac{1}{4}} - 4 \end{matrix}$
$\begin{matrix} {(a^{- \frac{1}{2}} + 3 + a^{\frac{1}{2}})}^{2} & = {(a^{- \frac{1}{2}})}^{2} + {(3)}^{2} + {(a^{\frac{1}{2}})}^{2} + 2 (a^{- \frac{1}{2}}) (3) + 2 (a^{- \frac{1}{2}}) (a^{\frac{1}{2}}) + 2 (3) (a^{\frac{1}{2}}) \\ = a^{- 1} + 9 + a + 6 a^{- \frac{1}{2}} + 2 a^{- \frac{1}{2} + \frac{1}{2}} + 6 a^{\frac{1}{2}} \\ = a^{- 1} + 9 + a + 6 a^{- \frac{1}{2}} + 2 + 6 a^{\frac{1}{2}} \\ = a^{- 1} + 6 a^{- \frac{1}{2}} + 11 + a + 6 a^{\frac{1}{2}} \end{matrix}$
$\begin{matrix} {(m + 2 m^{\frac{3}{4}} - 3 m^{\frac{1}{2}})}^{2} & = m^{2} + {(2 m^{\frac{3}{4}})}^{2} + {(- 3 m^{\frac{1}{2}})}^{2} + 2 m (2 m^{\frac{3}{4}}) + 2 m (- 3 m^{\frac{1}{2}}) + 2 (2 m^{\frac{3}{4}}) (- 3 m^{\frac{1}{2}}) \\ = m^{2} + 2^{2} m^{\frac{3}{2}} + {(- 3)}^{2} m + 2^{2} m^{1 + \frac{3}{4}} - 6 m^{1 + \frac{1}{2}} - 2^{2} .3 m^{\frac{3}{4} + \frac{1}{2}} \\ = m^{2} + 4 m^{\frac{3}{2}} + 9 m + 4 m^{\frac{7}{4}} - 6 m^{\frac{3}{2}} - 12 m^{\frac{5}{4}} \\ = m^{2} + 4 m^{\frac{7}{4}} - 2 m^{\frac{3}{2}} - 12 m^{\frac{5}{4}} + 9 m \end{matrix}$
$\begin{matrix} {(a^{\frac{1}{2}} b^{- \frac{1}{3}} - 2 + a^{- \frac{1}{2}} b^{\frac{1}{3}})}^{2} & = {(a^{\frac{1}{2}} b^{- \frac{1}{3}})}^{2} + {(- 2)}^{2} + {(a^{- \frac{1}{2}} b^{\frac{1}{3}})}^{2} + 2 (a^{\frac{1}{2}} b^{- \frac{1}{3}}) (- 2) + 2 (a^{\frac{1}{2}} b^{- \frac{1}{3}}) (a^{- \frac{1}{2}} b^{\frac{1}{3}}) + 2 (- 2) (a^{- \frac{1}{2}} b^{\frac{1}{3}}) \\ = a b^{- \frac{2}{3}} + 4 + a^{- 1} b^{\frac{2}{3}} - 4 a^{\frac{1}{2}} b^{- \frac{1}{3}} + 2 a^{\frac{1}{2} - \frac{1}{2}} b^{- \frac{1}{3} + \frac{1}{3}} - 4 a^{- \frac{1}{2}} b^{\frac{1}{3}} \\ = a b^{- \frac{2}{3}} + 4 + a^{- 1} b^{\frac{2}{3}} - 4 a^{\frac{1}{2}} b^{- \frac{1}{3}} + 2 - 4 a^{- \frac{1}{2}} b^{\frac{1}{3}} \\ = a^{- 1} b^{\frac{2}{3}} - 4 a^{- \frac{1}{2}} b^{\frac{1}{3}} + 6 - 4 a^{\frac{1}{2}} b^{- \frac{1}{3}} + a b^{- \frac{2}{3}} \end{matrix}$
$\begin{matrix} {(x^{\frac{1}{2}} + x^{\frac{1}{4}} - 1)}^{3} & = {(x^{\frac{1}{2}})}^{3} + {(x^{\frac{1}{4}})}^{3} + {(- 1)}^{3} + 3 {(x^{\frac{1}{2}})}^{2} (x^{\frac{1}{4}}) + 3 {(x^{\frac{1}{2}})}^{2} (- 1) + 3 {(x^{\frac{1}{4}})}^{2} (x^{\frac{1}{2}}) + 3 {(x^{\frac{1}{4}})}^{2} (- 1) + 3 {(- 1)}^{2} (x^{\frac{1}{4}}) + 3 {(- 1)}^{2} (x^{\frac{1}{2}}) \\ + 6 (- 1) (x^{\frac{1}{4}}) (x^{\frac{1}{2}}) \\ = x^{\frac{3}{2}} + x^{\frac{3}{4}} - 1 + 3 x x^{\frac{1}{4}} - 3 x + 3 x^{\frac{1}{2}} x^{\frac{1}{2}} - 3 x^{\frac{1}{2}} + 3 x^{\frac{1}{4}} + 3 x^{\frac{1}{2}} - 6 x^{\frac{1}{4}} x^{\frac{1}{2}} \\ = x^{\frac{3}{2}} + x^{\frac{3}{4}} - 1 + 3 x^{\frac{5}{4}} - 3 x + 3 x + 3 x^{\frac{1}{4}} - 6 x^{\frac{3}{4}} \\ = x^{\frac{3}{2}} + 3 x^{\frac{5}{4}} - 5 x^{\frac{3}{4}} + 3 x^{\frac{1}{4}} - 1 \end{matrix}$
$\begin{matrix} {(a^{\frac{2}{3}} - 2 + a^{- \frac{2}{3}})}^{3} & = {(a^{\frac{2}{3}})}^{3} + {(- 2)}^{3} + {(a^{- \frac{2}{3}})}^{3} + 3 {(a^{\frac{2}{3}})}^{2} (- 2) + 3 {(a^{\frac{2}{3}})}^{2} (a^{- \frac{2}{3}}) + 3 {(- 2)}^{2} (a^{\frac{2}{3}}) + 3 {(- 2)}^{2} (a^{- \frac{2}{3}}) + 3 {(a^{- \frac{2}{3}})}^{2} (a^{\frac{2}{3}}) + 3 {(a^{- \frac{2}{3}})}^{2} (- 2) \\ + 6 (a^{\frac{2}{3}}) (- 2) (a^{- \frac{2}{3}}) \\ = a^{2} - 8 + a^{- 2} - 6 a^{\frac{4}{3}} + 3 a^{\frac{4}{3}} a^{- \frac{2}{3}} + 3.4 a^{\frac{2}{3}} + 3.4 a^{- \frac{2}{3}} + 3 a^{- \frac{4}{3}} a^{\frac{2}{3}} - 6 a^{- \frac{4}{3}} - 12 a^{\frac{2}{3}} a^{- \frac{2}{3}} \\ = a^{2} - 8 + a^{- 2} - 6 a^{\frac{4}{3}} + 3 a^{\frac{2}{3}} + 12 a^{\frac{2}{3}} + 12 a^{- \frac{2}{3}} + 3 a^{- \frac{2}{3}} - 6 a^{- \frac{4}{3}} - 12 \\ = a^{2} - 6 a^{\frac{4}{3}} + 15 a^{\frac{2}{3}} - 20 + 15 a^{- \frac{2}{3}} - 6 a^{- \frac{4}{3}} + a^{- 2} \end{matrix}$
$\begin{matrix} {(m^{\frac{1}{6}} + 2 m^{\frac{1}{3}} + m^{\frac{1}{2}})}^{3} & = {(m^{\frac{1}{6}})}^{3} + {(2 m^{\frac{1}{3}})}^{3} + {(m^{\frac{1}{2}})}^{3} + 3 {(m^{\frac{1}{6}})}^{2} (2 m^{\frac{1}{3}}) + 3 {(m^{\frac{1}{6}})}^{2} (m^{\frac{1}{2}}) + 3 {(2 m^{\frac{1}{3}})}^{2} (m^{\frac{1}{6}}) + 3 {(2 m^{\frac{1}{3}})}^{2} (m^{\frac{1}{2}}) + 3 {(m^{\frac{1}{2}})}^{2} (m^{\frac{1}{6}}) \\ + 3 {(m^{\frac{1}{2}})}^{2} (2 m^{\frac{1}{3}}) + 6 m^{\frac{1}{6}} (2 m^{\frac{1}{3}}) (m^{\frac{1}{2}}) \\ = m^{\frac{1}{2}} + 2^{3} m + m^{\frac{3}{2}} + 6 m^{\frac{1}{3}} m^{\frac{1}{3}} + 3 m^{\frac{1}{3}} m^{\frac{1}{2}} + 3. 2^{2} m^{\frac{2}{3}} m^{\frac{1}{6}} + 3. 2^{2} m^{\frac{2}{3}} m^{\frac{1}{2}} + 3 m m^{\frac{1}{6}} + 6 m m^{\frac{1}{3}} + 12 m^{\frac{1}{6}} m^{\frac{1}{3}} m^{\frac{1}{2}} \\ = m^{\frac{1}{2}} + 8 m + m^{\frac{3}{2}} + 6 m^{\frac{2}{3}} + 3 m^{\frac{5}{6}} + 12 m^{\frac{5}{6}} + 12 m^{\frac{7}{6}} + 3 m^{\frac{7}{6}} + 6 m^{\frac{4}{3}} + 12 m \\ = m^{\frac{3}{2}} + 6 m^{\frac{4}{3}} + 15 m^{\frac{7}{6}} + 20 m + 15 m^{\frac{5}{6}} + 6 m^{\frac{2}{3}} + m^{\frac{1}{2}} \end{matrix}$

Categories: Capítulo XXX