Question

1) Simplificati fractiile urmatoare pana devin ireductibile:
a) 2 la puterea 19 supra 2 la puterea 21
b) 3 la puterea 34 supra 3 la puterea 37
c) 5 la puterea 63 supra 5 la puterea 65
d) 6 la puterea 47 supra 6 la puterea 48
e) 7 la puterea 78 supra 7 la puterea 80
f) 2 la puterea 65 supra 2 la puterea 70

2) Simplificati fractiile urmatoare
a)131313 supra 313131
b)171717 supra 717171
c)190019 supra 570057
d)50505 supra 373737
e)1515015 supra 5151051
f) 30303 supra 575757

Answer 1

Răspuns:

Explicație pas cu pas:

Folosim formula $\displaystyle\boxed{\frac{a^{n} }{a^{m} } =a^{n} : a^{m} =a^{n-m} }$  si  $\displaystyle\boxed{\ a^{-n} =\frac{1}{a^{n} }}$

a)

$\displaystyle\frac{2^{19} }{2^{21} } =2^{19-21} =2^{-2} =\frac{1}{2^{2} } =\frac{1}{4}$

b)

$\displaystyle\frac{3^{34} }{3^{37} } =3^{34-37} =3^{-3} =\frac{1}{3^{3} } =\frac{1}{27}$

c)

$\displaystyle \frac{5^{63} }{5^{65} } =5^{63-65} =5^{-2} =\frac{1}{5^{2} } =\frac{1}{25}$

d)

$\displaystyle\frac{6^{47} }{6^{48} } =6^{47-48} =6^{-1} =\frac{1}{6^{1} } =\frac{1}{6}$

f)

$\displaystyle\frac{2^{65} }{2^{70} } =2^{65-70} =2^{-5} =\frac{1}{2^{5} } =\frac{1}{32}$

Simplificarea fractiilor inseamna a gasi acel numar comun atat la numitor cat si la numarator adica daca $\displaystyle\boxed{\frac{a\times b}{a\times c} =\frac{b}{c} }$

a)

$\displaystyle\frac{131313}{313131} =\frac{3\times 7\times 13^{2}\times 37 }{3\times 7\times 31 \times 37} =\frac{\not3\times \not7\times 13^{2}\times \not37 }{\not3\times \not7\times 31\times \not37} =\frac{13^{2} }{31} =\frac{169}{31}$

b)

$\displaystyle\frac{171717}{717171} =\frac{3\times7\times13\times17\times37}{3\times7\times13\times37\times71} =\frac{\not3\times\not7\times\not13\times17\times\not37}{\not3\times\not7\times\not13\times\not37\times71} =\frac{17}{71}$

c)

$\displaystyle\frac{190019}{570057} =\frac{19\times73\times137}{3\times19\times73\times137} =\frac{\not19\times\not73\times\not137}{3\times\not19\times\not73\times\not137} =\frac{1}{3}$

d)

$\displaystyle\frac{50505}{373737} =\frac{3\times5\times7\times13\times37}{3\times7\times13\times37^{2} } =\frac{\not3\times5\times\not7\times\not13\times\not37}{\not3\times\not7\times\not13\times\not37^{2} } =\frac{5}{37}$

e)

$\displaystyle\frac{1515015}{5151051} =\frac{3^{2}\times5\times131\times257 }{3^{2}\times17\times131\times257 } =\frac{\not3^{2}\times5\times\not131\times\not257 }{\not3^{2}\times17\times\not131\times\not257 } =\frac{5}{17}$

f)

$\displaystyle\frac{30303}{575757} =\frac{3^{2}\times7\times13\times37 }{3^{2}\times7\times13\times19\times37 } =\frac{\not3^{2}\times\not7\times\not13\times\not37 }{\not3^{2}\times\not7\times\not13\times19\times\not37 } =\frac{1}{19}$