Question

1) Calculați:
A)2^18*12^18:(6^16*4^16)=
B)75^38:25^38-(3^19)^2=
C)26^27:13^27:2^5^2=
D)10^64:2^64-5^4^3=
2)Efectuați:
A) (3^4)^7:9^13=
B) (2^9)^2:4^9=
C) 8^24:16^18
D) 125^9:25^13=
E) 81^15:27^20=
F) 8^13^2:64^84=
G) 3^7^2:81^12=
3) Determinați numărul natural "n" pentru care au loc egalitățile:
A) 3 la puterea 2•n-3=27^7:9^2
B) 5 la puterea 3•n+1=25^11
C) 7 la puterea n+1:7^4=49^2^3
D) 10 la puterea 2•n+1:5 la puterea 2•n+1=8^9
4) Comparați numerele:
A) 8^17 cu 16^12;
B) 27^13 cu 9^20;
C) 3^27 cu 9^13;
D) 2^51 cu 3^34;
E) 4^115 cu 5^92.
5) Ordonați crescător numerele:
A) 2^35; 4^18; 8^11;
B) 3^61; 9^29; 27^20;
C) 125^14; 25^20; 5^39.
6) Arătați ca următoarele numere sunt pătrate perfecte:
A) 32*93-32*85=
B) 2^83-4^41=
C) 11*(5^21-5^20-5^18)=
Va rog sa mă ajutați repede! Mulțumesc anticipat!!

Answer 1

$1a).2^1^8*12^1^8:(6^1^6*4^1^6)=(2*12)^1^8:(6*4)^1^6=24^1^8:24^1^6= \\ =24^1^8^-^1^6=24^2=576 \\ b).75^3^8:25^3^8-(3^1^9)^2=(75:25)^3^8-3^3^8=3^3^8-3^3^8=0 \\ c).26^2^7:13^2^7: 2^{ 5^{2} } =(26:13)^2^7:2^2^5=2^2^ 7:2^2^5=2^2^7^-^2^5=2^2=4 \\ d).10^6^4:2^6^4- 5^{ 4^{3} } =(10:2)^6^4-5^6^4=5^6^4-5^6^4=0$
$2a).(3^4)^7:9^1^3=3^2^8:(3^2)^1^3=3^2^8:3^2^6=3^2^8^-^2^6=3^2=9 \\ b).(2^9)^2:4^9=2^1^8:(2^2)^9=2^1^8:2^1^8=1 \\ c).8^2^4:16^1^8=(2^3)^2^4:(2^4)^1^8=2^7^2:2^7^2=1 \\ d).125^9:25^1^3=(5^3)^9:(5^2)^1^3=5^2^7:5^2^6=5^2^7^-^2^5=5^1=5 \\ e).81^1^5:27^2^0=(3^4)^1^5:(3^3)^2^0=3^6^0:3^6^0=1 \\ f). 8^{ 13^{2} } :64^8^4=8^1^6^9:(8^2)^8^4=8^1^6^9:8^1^6^8=8^1^6^9^-^1^6^8=8^1=8 \\ g). 3^{ 7^{2} } :81^1^2=3^4^9:(3^4)^1^2=3^4^9:3^4^8=3^4^9^-^4^8=3^1=3$
$3a).3^2^n^-^3=26^7:9^2 \\ 3^2^n^-^3=3^6^3:3^4 \\ 3^2^n^-^3=3^5^9 \\ 2n-3=59 \\ 2n=59+3 \\ 2n=62 \\ n=62:2 \\ n=31$
$b).5^3^n^+^1=25^1^1 \\ 5^3^n^+^1=5^2^2 \\ 3n+1=22 \\ 3n=22-1 \\ 3n=21 \\ n=21:3 \\ n=7$
$c).7^n^+^1:7^4= 49^{ 2^{3} } \\ 7^n^+^1:7^4=49^8 \\ 7^n^+^1^-^4=7^1^6 \\ 7^n^-^3=7^1^6 \\ n-3=16 \\ n=16+3 \\ n=19$
$10^2^n^+^1:5^2^n^+^1=8^9 \\ 10^2^n^+^1:5^2^n^+^1=(2^3)^9 \\ 2^2^n^+^1=2^2^7 \\ 2n+1=27 \\ 2n=27-1 \\ 2n=26 \\ n=26:2 \\ n=13$
$4a).8^1^7;16^1^2 \\ 8^1^7=(2^3)^1^7=2^5^1;16^1^2=(2^4)^1^2=2^4^8 \\ 8^1^7\ \textgreater \ 16^1^2 \\ b).27^1^3;9^2^0 \\ 27^1^3=(3^3)^1^3=3^3^9;9^2^0=(3^2)^2^0=3^4^0 \\ 27^1^3\ \textless \ 9^2^0 \\ c).3^2^7;9^1^3 \\ 3^2^7;9^1^3=(3^2)^1^3=3^2^6 \\ 3^2^7\ \textgreater \ 9^1^3$
$d).2^5^1;3^3^4 \\ 2^5^1\ \textgreater \ 3^3^4 \\ e).4^1^1^5;5^9^2 \\ 4^1^1^5\ \textless \ 5^9^2$
$5a).2^3^5;4^1^8;8^1^1 \\ 2^3^5;4^1^8=(2^2)^1^8=2^3^6;8^1^1=(2^3)^1^1=2^3^3 \\ 8^1^1;2^3^5;4^1^8 \\ b).3^6^1;9^2^9;27^2^0 \\ 3^6^1;9^2^9=(3^2)^2^9=3^5^8;27^2^0=(3^3)^2^0=3^6^0 \\ 9^2^9 ;27^2^0;3^6^1 \\ c).125^1^4;25^2^0;5^3^9 \\ 125^1^4=(5^3)^1^4=5^4^2;25^2^0=(5^2)^2^0=5^4^0;5^3^9 \\ 5^3^9;25^2^0;125^1^4$
$6a)32*93-32*85=32(93-85)=32*8=2^ 5*2^3=2^8=(2^4)^2=16^2$
$b).2^8^3-4^4^1=2^8^3-(2^2)^4^1=2^8^3-2^8^2=2^8^2(2-1)=2^8^2=(2^4^1)^2$
$c).11*(5^2^1-5^2^0-5^1^8)=11*5^1^8(5^3-5^2-1)= \\ =11-5^1^8(125-25-1)=11*5^1^8*99= \\ =11*11*9*5^1^8=11^2*3^2*(5^9)^2$