[tex]c)\\\\\dfrac{5^1^3\cdot3+2\cdot5^1^3}{(5^1^1:25^4)\cdot25}=\dfrac{3\cdot5^1^3+2\cdot5^1^3}{((5^1^1:(5^2)^4)\cdot5^2}=\dfrac{(3+2)\cdot5^1^3}{(5^1^1:5^8)\cdot5^2}=\dfrac{5\cdot5^1^3}{5^{11-8}\cdot5^2}=\dfrac{5^{1+13}}{5^3\cdot5^2}=\\\\=\dfrac{5^1^4}{5^5} =5^{14-5}=5^9[/tex]
[tex]d)\\\\\dfrac{(9^3)^2\cdot27^2}{3^1^7-2\cdot3^1^6}=\dfrac{9^6\cdot(3^3)^2}{3\cdot3^1^6-2\cdot3^1^6}=\dfrac{(3^2)^6\cdot3^6}{(3-2)\cdot3^1^6}=\dfrac{3^1^2\cdot3^6}{1\cdot3^1^6}=\dfrac{3^{12+6}}{3^1^6}=\dfrac{3^1^8}{3^1^6}=\\\\\\=3^{18-16}=3^2=9[/tex]
[tex]e)\\\\\dfrac{4\cdot6^1^4-6^1^5}{3^1^3\cdot2^1^5+2^1^5\cdot3^1^3}=\dfrac{4\cdot6^1^4-6\cdot6^1^4}{3^1^3\cdot2^1^3\cdot2^2+2^2\cdot2^1^3\cdot3^1^3}=\dfrac{(4-6)\cdot6^1^4}{(3\cdot2)^1^3\cdot2^2+2^2\cdot(2\cdot3)^1^3}=\\\\=\dfrac{-2\cdot6^1^4}{6^1^3\cdot2^2+2^2\cdot6^1^3}=\dfrac{-2\cdot6^1^4}{2\cdot2^2\cdot6^1^3}=\dfrac{-2\cdot\not6^1^4}{2\cdot4\cdot\not6^1^3}=\dfrac{-2\cdot6^{14-13}}{8}=\\\\\\=\dfrac{-2\cdot6^1}{8}=\dfrac{-2\cdot6}{8}=\dfrac{-12}{8}=-\dfrac{3}{2}[/tex]
[tex]f)\\\\\\\dfrac{10^9-(2^4\cdot5^5)^2}{1500:0,001}=\dfrac{10^9-(2^4\cdot5^4\cdot5)^2}{1500000}=\dfrac{10^9-((2\cdot5)^4\cdot5)^2}{1500000}=\dfrac{10^9-(10^4\cdot5)^2}{1500000}=\\\\=\dfrac{10^9-(10^4)^2\cdot5^2}{1500000}=\dfrac{10^9-10^8\cdot5^2}{1500000}=\dfrac{10\cdot10^8-10^8\cdot5^2}{1500000}=\dfrac{(10-1)\cdot10^8}{1500000}=\\\\\\=\dfrac{9\cdot10^8}{1500000}=\dfrac{9\cdot100000000}{1500000}=\dfrac{900000000}{1500000}=600[/tex]
Zastosowano wzory
[tex](a^m)^n=a^{m\cdot n}\\\\a^m\cdot a^n=a^{m+n}\\\\\frac{a^m}{a^n}=a^{m-n}\\\\a^n\cdot b^n=(a\cdotb)^n[/tex]
Odpowiedź:
b)
[4^26 : (4² * 4)^4]/(9 * 2^20 + 7 * 2^20) = [4^26 : (4³)^4]/[2^20(9 + 7)] =
= (4^26 : 4^12)/(2^20 * 16) = 4^14/(2^20 * 2^4) = 4^14 : 2^24 = (2²)^14 : 2^24 =
= 2^28 : 2^24 = 2^4 = 16
c)
(5^13 * 3 + 2 * 5^13)/[(5^11 : 25^4)³ * 25] = [5^13(3 + 2)]/{[5^11 : (5²)^4]³ * 5²} =
= (5^13 * 5)/[(5^11 : 5^8)³ * 5²] = 5^14/[(5³)³ * 5²] = 5^14/(5^9 * 5²) =
= 5^14 : 5^11 = 5³= 125
e)
(4 * 6^14 - 6^15/(3^13 * 2^15 + 2^15 * 3^13) = [6^14(4 - 6)/(2 * 3^13 * 2^15) =
= -2 * 6^14/(3^13 * 2^16) = -2 * 6^14/[2³(3^13 * 2^13) = -2 * 6^14/(2³ * 6^13) =
= -2^-2 * 6 = -(1/2)² * 6 = -1/4 * 6 = -1/2 * 3 = -3/2 = -1 1/2
f)
[10^9 - (2^4 * 5^5)²]/(1500 : 0,001) = (10^9 - 2^8 * 5^10)/(1500 : 0,001) =
= (10^9 - 5² * 10^8)/1 500 000 = 10^8(10 - 25)/(1,5 * 10^6) =
= 10^8(-15)/(1,5 * 10^6) = 10²(-1,5 * 10)/1,5 = -10³ = -1000
11.
a)
(7^12 * 3 + 4 * 7^12)/7^10 = 7^12(3 + 4)/7^10 = 7^12 * 7 : 7^10 = 7^13 : 7^10 =
= 7³ = 343
d)
[(9³)² * 27²]/(3^17 - 2 * 3^16) = [9^6 * (3³)²]/[3^16(3 - 2)] =
= [(3²)^6 * 3^6]/3^16 = (3^12 * 3^6)/3^16 = 3^18/3^16 = 3² = 9
Szczegółowe wyjaśnienie:
" Life is not a problem to be solved but a reality to be experienced! "
© Copyright 2013 - 2024 KUDO.TIPS - All rights reserved.
[tex]c)\\\\\dfrac{5^1^3\cdot3+2\cdot5^1^3}{(5^1^1:25^4)\cdot25}=\dfrac{3\cdot5^1^3+2\cdot5^1^3}{((5^1^1:(5^2)^4)\cdot5^2}=\dfrac{(3+2)\cdot5^1^3}{(5^1^1:5^8)\cdot5^2}=\dfrac{5\cdot5^1^3}{5^{11-8}\cdot5^2}=\dfrac{5^{1+13}}{5^3\cdot5^2}=\\\\=\dfrac{5^1^4}{5^5} =5^{14-5}=5^9[/tex]
[tex]d)\\\\\dfrac{(9^3)^2\cdot27^2}{3^1^7-2\cdot3^1^6}=\dfrac{9^6\cdot(3^3)^2}{3\cdot3^1^6-2\cdot3^1^6}=\dfrac{(3^2)^6\cdot3^6}{(3-2)\cdot3^1^6}=\dfrac{3^1^2\cdot3^6}{1\cdot3^1^6}=\dfrac{3^{12+6}}{3^1^6}=\dfrac{3^1^8}{3^1^6}=\\\\\\=3^{18-16}=3^2=9[/tex]
[tex]e)\\\\\dfrac{4\cdot6^1^4-6^1^5}{3^1^3\cdot2^1^5+2^1^5\cdot3^1^3}=\dfrac{4\cdot6^1^4-6\cdot6^1^4}{3^1^3\cdot2^1^3\cdot2^2+2^2\cdot2^1^3\cdot3^1^3}=\dfrac{(4-6)\cdot6^1^4}{(3\cdot2)^1^3\cdot2^2+2^2\cdot(2\cdot3)^1^3}=\\\\=\dfrac{-2\cdot6^1^4}{6^1^3\cdot2^2+2^2\cdot6^1^3}=\dfrac{-2\cdot6^1^4}{2\cdot2^2\cdot6^1^3}=\dfrac{-2\cdot\not6^1^4}{2\cdot4\cdot\not6^1^3}=\dfrac{-2\cdot6^{14-13}}{8}=\\\\\\=\dfrac{-2\cdot6^1}{8}=\dfrac{-2\cdot6}{8}=\dfrac{-12}{8}=-\dfrac{3}{2}[/tex]
[tex]f)\\\\\\\dfrac{10^9-(2^4\cdot5^5)^2}{1500:0,001}=\dfrac{10^9-(2^4\cdot5^4\cdot5)^2}{1500000}=\dfrac{10^9-((2\cdot5)^4\cdot5)^2}{1500000}=\dfrac{10^9-(10^4\cdot5)^2}{1500000}=\\\\=\dfrac{10^9-(10^4)^2\cdot5^2}{1500000}=\dfrac{10^9-10^8\cdot5^2}{1500000}=\dfrac{10\cdot10^8-10^8\cdot5^2}{1500000}=\dfrac{(10-1)\cdot10^8}{1500000}=\\\\\\=\dfrac{9\cdot10^8}{1500000}=\dfrac{9\cdot100000000}{1500000}=\dfrac{900000000}{1500000}=600[/tex]
Zastosowano wzory
[tex](a^m)^n=a^{m\cdot n}\\\\a^m\cdot a^n=a^{m+n}\\\\\frac{a^m}{a^n}=a^{m-n}\\\\a^n\cdot b^n=(a\cdotb)^n[/tex]
Odpowiedź:
b)
[4^26 : (4² * 4)^4]/(9 * 2^20 + 7 * 2^20) = [4^26 : (4³)^4]/[2^20(9 + 7)] =
= (4^26 : 4^12)/(2^20 * 16) = 4^14/(2^20 * 2^4) = 4^14 : 2^24 = (2²)^14 : 2^24 =
= 2^28 : 2^24 = 2^4 = 16
c)
(5^13 * 3 + 2 * 5^13)/[(5^11 : 25^4)³ * 25] = [5^13(3 + 2)]/{[5^11 : (5²)^4]³ * 5²} =
= (5^13 * 5)/[(5^11 : 5^8)³ * 5²] = 5^14/[(5³)³ * 5²] = 5^14/(5^9 * 5²) =
= 5^14 : 5^11 = 5³= 125
e)
(4 * 6^14 - 6^15/(3^13 * 2^15 + 2^15 * 3^13) = [6^14(4 - 6)/(2 * 3^13 * 2^15) =
= -2 * 6^14/(3^13 * 2^16) = -2 * 6^14/[2³(3^13 * 2^13) = -2 * 6^14/(2³ * 6^13) =
= -2^-2 * 6 = -(1/2)² * 6 = -1/4 * 6 = -1/2 * 3 = -3/2 = -1 1/2
f)
[10^9 - (2^4 * 5^5)²]/(1500 : 0,001) = (10^9 - 2^8 * 5^10)/(1500 : 0,001) =
= (10^9 - 5² * 10^8)/1 500 000 = 10^8(10 - 25)/(1,5 * 10^6) =
= 10^8(-15)/(1,5 * 10^6) = 10²(-1,5 * 10)/1,5 = -10³ = -1000
11.
a)
(7^12 * 3 + 4 * 7^12)/7^10 = 7^12(3 + 4)/7^10 = 7^12 * 7 : 7^10 = 7^13 : 7^10 =
= 7³ = 343
b)
[4^26 : (4² * 4)^4]/(9 * 2^20 + 7 * 2^20) = [4^26 : (4³)^4]/[2^20(9 + 7)] =
= (4^26 : 4^12)/(2^20 * 16) = 4^14/(2^20 * 2^4) = 4^14 : 2^24 = (2²)^14 : 2^24 =
= 2^28 : 2^24 = 2^4 = 16
d)
[(9³)² * 27²]/(3^17 - 2 * 3^16) = [9^6 * (3³)²]/[3^16(3 - 2)] =
= [(3²)^6 * 3^6]/3^16 = (3^12 * 3^6)/3^16 = 3^18/3^16 = 3² = 9
e)
(4 * 6^14 - 6^15/(3^13 * 2^15 + 2^15 * 3^13) = [6^14(4 - 6)/(2 * 3^13 * 2^15) =
= -2 * 6^14/(3^13 * 2^16) = -2 * 6^14/[2³(3^13 * 2^13) = -2 * 6^14/(2³ * 6^13) =
= -2^-2 * 6 = -(1/2)² * 6 = -1/4 * 6 = -1/2 * 3 = -3/2 = -1 1/2
Szczegółowe wyjaśnienie: