Question

1. Encontrar las potencias, raíces y
logaritmos de los siguientes
ejercicios​

Answer 1

Las potencias, raíces y logaritmos de los ejercicios son:

a. $(\frac{5}{3})^{4} =\frac{625}{81}$

b. $(\frac{7}{2})^{3} =\frac{343}{8}$

c. tex](\frac{6}{9})^{6} =\frac{64}{729} [/tex]

d. $\sqrt[3]{\frac{27}{125}} =\frac{3}{5}$

e. $\sqrt[3]{\frac{27}{125}} =\frac{5}{2}$

f. $\sqrt[4]{\frac{625}{16}}= \frac{5}{2}$

g. $log_{3/35}(\frac{27}{125})=3.log_{3/35}(3)-3log_{3/35}(5)$

h. $log_{1/7}(\frac{1}{49})=2$

a. $(\frac{5}{3})^{4}$

Aplicamos propiedades de exponente: $(\frac{a}{b})^{c}=\frac{a^{c}}{b^{c}}$

$(\frac{5}{3})^{4}= \frac{5^{4}}{3^{4}}$

$(\frac{5}{3})^{4} =\frac{625}{81}$

b. $(-\frac{7}{2})^{3}$

Aplicamos propiedades de exponente: $(\frac{a}{b})^{c}=\frac{a^{c}}{b^{c}}$

$(-\frac{7}{2})^{3}= -\frac{7^{3}}{2^{3}}$

$(-\frac{7}{2})^{3} = -\frac{343}{8}$

c. $(-\frac{6}{9})^{6}$

Aplicamos propiedades de exponente: $(\frac{a}{b})^{c}=\frac{a^{c}}{b^{c}}$

$(\frac{6}{9})^{6}= \frac{6^{6}}{9^{6}}$

6 y 9 son múltiplos de 3;

6/9 = 2/3

$(\frac{6}{9})^{6}= \frac{2^{6}}{3^{6}}$

$(\frac{6}{9})^{6} =\frac{64}{729}$

d. $\sqrt[3]{\frac{27}{125} }$

Reescribimos;

27 = 3³

125 = 5³

$\sqrt[3]{\frac{27}{125}} =\sqrt[3]{\frac{3^{3}}{5^{3}}}$

Aplicamos propiedades de raíces: $\sqrt[n]{\frac{a}{b}}= \frac{\sqrt[n]{a} }{\sqrt[n]{b} }$

$\sqrt[3]{\frac{27}{125}}= \frac{\sqrt[3]{3^{3}} }{\sqrt[3]{5^{3}} }$

$\sqrt[3]{\frac{27}{125}} =\frac{3}{5}$

e.  $\sqrt[4]{\frac{625}{16} }$

Reescribimos;

625 = 25²

16 = 4²

$\sqrt[4]{\frac{625}{16}} =\sqrt[4]{\frac{25^{2}}{4^{2}}}$

Aplicamos propiedades de raíces: $\sqrt[n]{\frac{a}{b}}= \frac{\sqrt[n]{a} }{\sqrt[n]{b} }$

$\sqrt[4]{\frac{625}{16}}= \frac{\sqrt[4]{25^{2}} }{\sqrt[4]{4^{2}} }$

$\sqrt[4]{\frac{625}{16}}= \frac{\sqrt{25}}{\sqrt{4}}$

$\sqrt[3]{\frac{27}{125}} =\frac{5}{2}$

f.  $\sqrt[5]{\frac{1}{32} }$

Reescribimos;

32 = 2^5

$\sqrt[5]{\frac{1}{32}} =\sqrt[5]{\frac{1}{2^{5}}}$

Aplicamos propiedades de raíces: $\sqrt[n]{\frac{a}{b}}= \frac{\sqrt[n]{a} }{\sqrt[n]{b} }$

$\sqrt[5]{\frac{1}{32}}= \frac{\sqrt[5]{1} }{\sqrt[5]{2^{5}} }$

$\sqrt[4]{\frac{625}{16}}= \frac{5}{2}$

g. $log_{3/35}(\frac{27}{125})$

Aplicar propiedad de logaritmos: $log_{c}(\frac{a}{b})=log_{c}(a)-log_{c}(b)$

$log_{3/35}(\frac{27}{125})=log_{3/35}(27)-log_{3/35}(125)$

Reescribir;

27 = 3³

125 = 5³

$log_{3/35}(\frac{27}{125})=log_{3/35}(3^{3})-log_{3/35}(5^{3})$

Aplicando propiedades de logaritmos: $log_{a} (x^{b}) = b.log_{a}(x)$

$log_{3/35}(\frac{27}{125})=3.log_{3/35}(3)-3log_{3/35}(5)$

h. $log_{1/7}(\frac{1}{49})$

Reescribimos:

1/49 = (1/7)²

$log_{1/7}(\frac{1}{49})=log_{1/7}( (\frac{1}{7})^{2})$

Aplicando propiedades de logaritmos: $log_{a} (x^{b}) = b.log_{a}(x)$

$log_{1/7}(\frac{1}{49})=2.log_{1/7}(\frac{1}{7})$

$log_{1/7}(\frac{1}{49})=2$