Operaciones combinadas con fracciones. Operaciones con potencias. (1)

Pregunta 1 [3 puntos]

Realizar las siguientes operaciones combinadas con fracciones:

\(\displaystyle\frac{\displaystyle\frac{2}{3}}{\displaystyle\frac{8}{9}}-\frac{\displaystyle\frac{2}{3}-\frac{3}{5}}{1+\displaystyle\frac{1}{3}}\)
\(\displaystyle\frac{\displaystyle\frac{1}{4}+\frac{1}{3}}{\displaystyle\frac{3}{5}-\frac{1}{4}}\cdot\frac{2}{5}-1\)

Las solución aquí

\(\displaystyle\frac{\displaystyle\frac{2}{3}}{\displaystyle\frac{8}{9}}-\frac{\displaystyle\frac{2}{3}-\frac{3}{5}}{1+\displaystyle\frac{1}{3}}=\)

\(=\displaystyle\frac{18}{24}-\frac{\displaystyle\frac{10}{15}-\frac{9}{15}}{\displaystyle\frac{3}{3}+\frac{1}{3}}=\frac{18}{24}-\frac{\displaystyle\frac{1}{15}}{\displaystyle\frac{4}{3}}=\)

\(=\displaystyle\frac{18}{24}-\frac{3}{60}=\frac{90}{120}-\frac{6}{120}=\frac{84}{120}=\frac{7}{10}\)

\(\displaystyle\frac{\displaystyle\frac{1}{4}+\frac{1}{3}}{\displaystyle\frac{3}{5}-\frac{1}{4}}\cdot\frac{2}{5}-1=\)

\(=\displaystyle\frac{\displaystyle\frac{3}{12}+\frac{4}{12}}{\displaystyle\frac{12}{20}-\frac{5}{20}}\cdot\frac{2}{5}-1=\frac{\displaystyle\frac{7}{12}}{\displaystyle\frac{7}{20}}\cdot\frac{2}{5}-1=\)

\(=\displaystyle\frac{140}{84}\cdot\frac{2}{5}-1=\frac{280}{420}-1=\frac{280}{420}-\frac{420}{420}=-\frac{140}{420}=-\frac{1}{3}\)

Pregunta 2 [2 puntos]

Calcula:

\(3^3-(-4)^2+2^4+(-2)^3-5^0+6^1\)
\((-6)^2-(-3)^3+(-2)^5-9^2+(-5)^1\)

Las solución aquí

\(3^3-(-4)^2+2^4+(-2)^3-5^0+6^1=\)

\(=27-16+16+(-8)-1+6=\)

\(=27-16+16-8-1+6=24\)

\((-6)^2-(-3)^3+(-2)^5-9^2+(-5)^1=\)

\(=36-(-27)+(-32)-81+(-5)=\)

\(=36+27-32-81-5=-55\)

Pregunta 3 [2 puntos]

Efectúa las siguientes operaciones simplificando previamente las fracciones:

\(\displaystyle\left[\left(\frac{4}{2}\cdot\frac{2}{5}\right)^2:\left(\frac{3}{4}\cdot\frac{2}{5}\cdot\frac{5}{3}\right)^3\right]:\left[\left(3\cdot4^2-(-5)^2\cdot2\right)\cdot\left(\frac{2}{5}:\frac{4}{20}\right)\right]\)
\(\displaystyle\left(\frac{3}{4}-\frac{1}{8}-\frac{7}{24}\right)^2\cdot\left(\frac{1}{2}:\frac{1}{6}\right)^3\)

Las solución aquí

\(\displaystyle\left[\left(\frac{4}{2}\cdot\frac{2}{5}\right)^2:\left(\frac{3}{4}\cdot\frac{2}{5}\cdot\frac{5}{3}\right)^3\right]:\left[\left(3\cdot4^2-(-5)^2\cdot2\right)\cdot\left(\frac{2}{5}:\frac{4}{20}\right)\right]=\)

\(\displaystyle=\left[\left(\frac{8}{10}\right)^2:\left(\frac{30}{60}\right)^3\right]:\left[(48-50)\cdot\left(\frac{40}{20}\right)\right]=\left[\left(\frac{4}{5}\right)^2:\left(\frac{1}{2}\right)^3\right]:\left[(-2)\cdot2\right]=\)

\(=\displaystyle\left(\frac{16}{25}:\frac{1}{8}\right):(-4)=\frac{128}{25}:(-4)=\frac{128}{-100}=-\frac{32}{25}\)

\(\displaystyle\left(\frac{3}{4}-\frac{1}{8}-\frac{7}{24}\right)^2\cdot\left(\frac{1}{2}:\frac{1}{6}\right)^3=\)

\(=\displaystyle\left(\frac{18}{24}-\frac{3}{24}-\frac{7}{24}\right)^2\cdot\left(\frac{6}{2}\right)^3=\left(\frac{8}{24}\right)^2\cdot3^3=\)

\(=\displaystyle\left(\frac{1}{3}\right)^2\cdot3^3=\frac{1}{3^2}\cdot3^3=\frac{1}{9}\cdot27=\frac{27}{9}=3\)

Pregunta 4 [1 punto]

Expresa en forma de una sola potencia las siguientes expresiones:

\(\displaystyle(-3)^2\cdot3^{-5}\cdot(-3)^8\)
\(\displaystyle\left(-\frac{2}{5}\right)^4:\left(\frac{2}{5}\right)^{-3}\)

Las solución aquí

\(\displaystyle(-3)^2\cdot3^{-5}\cdot(-3)^8=\)

\(=3^2\cdot3^{-5}\cdot3^8=3^{2+(-5)+8}=3^5\)

\(\displaystyle\left(-\frac{2}{5}\right)^4:\left(\frac{2}{5}\right)^{-3}=\)

\(\displaystyle=\left(\frac{2}{5}\right)^4:\left(\frac{2}{5}\right)^{-3}=\left(\frac{2}{5}\right)^{4-(-3)}=\left(\frac{2}{5}\right)^7\)

Pregunta 5 [2 puntos]

Calcula:

\(\displaystyle\left(\frac{12}{4}\right)^2-2^3\cdot2^{-1}+\frac{3^2\cdot3^{-3}\cdot3^6}{3\cdot3^2}+5^2\cdot5^{-3}\cdot5^3\)
\(\displaystyle \frac{5^3\cdot5^4\cdot5^{-9}\cdot5^7}{5^6\cdot5^{-3}}+5^3-2\cdot2^0\cdot2^3-\frac{5^5}{5^4}\)