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B1: n2 + 6n + 8 = n2 + 4n + 2n + 8 = n(n+4) + 2(n+4) = (n+2)(n+4)
Vì n+2 < n+4 => n + 2 = 1 => n = -1
=> A = 3 nguyên tố, thoả
B2: x + y + xy = 2
=> x(y+1) + (y+1) = 3
=> (x+1)(y+1) = 3
Ta có:
x+1 | 1 | 3 | -1 | -3 |
y+1 | 3 | 1 | -3 | -1 |
x | 0 | 2 | -2 | -4 |
y | 2 | 0 | -4 | -2 |
Vậy (x,y) = .....................
B3: a : b = c dư r
=> 112 : b = 5 dư r
=> 112 : 5 = b dư r
=> 112 - r chia hết cho 5 và r < 5
=> r = 2 => b = 22
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Bài 6 :
a) \(\dfrac{625}{5^n}=5\Rightarrow\dfrac{5^4}{5^n}=5\Rightarrow5^{4-n}=5^1\Rightarrow4-n=1\Rightarrow n=3\)
b) \(\dfrac{\left(-3\right)^n}{27}=-9\Rightarrow\dfrac{\left(-3\right)^n}{\left(-3\right)^3}=\left(-3\right)^2\Rightarrow\left(-3\right)^{n-3}=\left(-3\right)^2\Rightarrow n-3=2\Rightarrow n=5\)
c) \(3^n.2^n=36\Rightarrow\left(2.3\right)^n=6^2\Rightarrow\left(6\right)^n=6^2\Rightarrow n=6\)
d) \(25^{2n}:5^n=125^2\Rightarrow\left(5^2\right)^{2n}:5^n=\left(5^3\right)^2\Rightarrow5^{4n}:5^n=5^6\Rightarrow\Rightarrow5^{3n}=5^6\Rightarrow3n=6\Rightarrow n=3\)
Bài 7 :
a) \(3^x+3^{x+2}=9^{17}+27^{12}\)
\(\Rightarrow3^x\left(1+3^2\right)=\left(3^2\right)^{17}+\left(3^3\right)^{12}\)
\(\Rightarrow10.3^x=3^{34}+3^{36}\)
\(\Rightarrow10.3^x=3^{34}\left(1+3^2\right)=10.3^{34}\)
\(\Rightarrow3^x=3^{34}\Rightarrow x=34\)
b) \(5^{x+1}-5^x=100.25^{29}\Rightarrow5^x\left(5-1\right)=4.5^2.\left(5^2\right)^{29}\)
\(\Rightarrow4.5^x=4.25^{2.29+2}=4.5^{60}\)
\(\Rightarrow5^x=5^{60}\Rightarrow x=60\)
c) Bài C bạn xem lại đề
d) \(\dfrac{3}{2.4^x}+\dfrac{5}{3.4^{x+2}}=\dfrac{3}{2.4^8}+\dfrac{5}{3.4^{10}}\)
\(\Rightarrow\dfrac{3}{2.4^x}-\dfrac{3}{2.4^8}+\dfrac{5}{3.4^{x+2}}-\dfrac{5}{3.4^{10}}=0\)
\(\Rightarrow\dfrac{3}{2}\left(\dfrac{1}{4^x}-\dfrac{1}{4^8}\right)+\dfrac{5}{3.4^2}\left(\dfrac{1}{4^x}-\dfrac{1}{4^8}\right)=0\)
\(\Rightarrow\left(\dfrac{1}{4^x}-\dfrac{1}{4^8}\right)\left(\dfrac{3}{2}+\dfrac{5}{3.4^2}\right)=0\)
\(\Rightarrow\dfrac{1}{4^x}-\dfrac{1}{4^8}=0\)
\(\Rightarrow\dfrac{4^8-4^x}{4^{x+8}}=0\Rightarrow4^8-4^x=0\left(4^{x+8}>0\right)\Rightarrow4^x=4^8\Rightarrow x=8\)
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\(\left(\frac{1}{0,125}\right)^n=128\Rightarrow\left(\frac{1}{\frac{1}{8}}\right)^n=128\Rightarrow8^n=128\Rightarrow2^{3n}=2^7\Rightarrow3n=7\Rightarrow n=\frac{7}{3}\)
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\(\frac{1}{9}\times27^n=3^n\)
\(\frac{1}{9}=\frac{3^n}{27^n}\)
\(\frac{1}{9}=\left(\frac{1}{9}\right)^n\)
\(\Rightarrow n=1\)
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(n^2 - 3) . (n^2 - 36) < 0
<=> n^2 - 3 và n^2 - 36 trái dấu
<=> n^2 - 3 > 0 ; n^2 - 36 < 0 hoặc n^2 - 3 < 0 ; n^2 - 36 > 0
<=> n > \(\sqrt{3}\) ; n < 6 hoặc n < \(\sqrt{3}\) ; n > 6 (loại)
Vậy \(\sqrt{3}\) < n < 6 thỏa mãn