21-6
Chứng minh rằng : a, M = 21^9+21^8+21^7 +....+ 21+1 chia hết cho 2 và 5 b, N = 6+6^2+6^3 +....+ 6^2020 chia hết cho 7 nhưng không chia hết cho 9 c, P = 4+4^2+4^3 +....+ 4^23+4^24 chia hết cho 20 và 21 d, Q = 6+6^2+6^3 +....+ 6^99 chia hết cho 43
Hộ mình làm bài này nhá :))))))))
Giải:
a) \(M=21^9+21^8+21^7+...+21+1\)
Do \(21^n\) luôn có tận cùng là 1
\(\Rightarrow M=21^9+21^8+21^7+...+21+1\)
Tân cùng của M là:
\(1+1+1+1+1+1+1+1+1+1=10\) tận cùng là 0
\(\Rightarrow M⋮10\)
\(\Leftrightarrow M⋮2;5\)
b) \(N=6+6^2+6^3+...+6^{2020}\)
\(N=6.\left(1+6\right)+6^3.\left(1+6\right)+...+6^{2019}.\left(1+6\right)\)
\(N=6.7+6^3.7+...+6^{2019}.7\)
\(N=7.\left(6+6^3+...+6^{2019}\right)⋮7\)
\(\Rightarrow N⋮7\)
Ta thấy: \(N=6+6^2+6^3+...+6^{2020}⋮6\)
Mà \(6⋮̸9\)
\(\Rightarrow N⋮̸9\)
c) \(P=4+4^2+4^3+...+4^{23}+4^{24}\)
\(P=1.\left(4+4^2\right)+4^2.\left(4+4^2\right)+...+4^{20}.\left(4+4^2\right)+4^{22}.\left(4+4^2\right)\)
\(P=1.20+4^2.20+...+4^{20}.20+4^{22}.20\)
\(P=20.\left(1+4^2+...+4^{20}+4^{22}\right)⋮20\)
\(\Rightarrow P⋮20\)
\(P=4+4^2+4^3+...+4^{23}+4^{24}\)
\(P=4.\left(1+4+4^2\right)+...+4^{22}.\left(1+4+4^2\right)\)
\(P=4.21+...+4^{22}.21\)
\(P=21.\left(4+...+4^{22}\right)⋮21\)
\(\Rightarrow P⋮21\)
d) \(Q=6+6^2+6^3+...+6^{99}\)
\(Q=6.\left(1+6+6^2\right)+...+6^{97}.\left(1+6+6^2\right)\)
\(Q=6.43+...+6^{97}.43\)
\(Q=43.\left(6+...+6^{97}\right)⋮43\)
\(\Rightarrow Q⋮43\)
Chúc bạn học tốt!
bài 3: Tính:
1) 1/6 - -5/6 2) 6/13 - -14/39 3) 4/5 - 4/-18 4) 7/21 - 9/-36
5) -12/18 - -21/35 6) -3/21 - 6/42 7) -18/24 - 15/21 8) 1/6 - 2/5
các bn giup mik mik vs
1)1/6 - - 5/6 = 1/6 + 5/6 = 1
2)6/13 - -14/39 = 6/13 + 14/39= 32/39
3)4/5 - 4/-18 = 4/5 + 4/18= 46/45
4)7/21 - 9/-36 = 7/21 + 9/36 = 7/12
5)-12/18 - -21/35 = -12/18 + 21/35 = -1/15
6)-3/21 - 6/42 = -2/7
7)-18/24 - 15/21 = -41/28
8)1/6 - 2/5 =-7/30
|-2| + 6 =
(-120) + (-60) =
| 21 - ( - 7) | + ( - 6)=
( - 5) + 21 =
( - 70) + ( - 21) =
6 + | ( - 7) + ( - 6)| =
27 + -( 5) =
27 + ( - 5 )
(-2) + 6 = 4
(-120) + (-60) = -180
| 21 - ( - 7) | + ( - 6)= 23
( - 5) + 21 = 16
( - 70) + ( - 21) = -91
6 + | ( - 7) + ( - 6)| = 19
27 + -( 5) = 22
27 + ( - 5 ) = 22
Đúng thì k còn sai thì thui chứ đừng k sai nhé.
(-2)+6=4
(-120)+(-60)=-180
(-5)+21=16
(-70)+(-21)=-91
27+-(5)=22
27+(-5)=22
Tính:
\(\sqrt{21+6\sqrt{6}}-\sqrt{21-6\sqrt{6}}\)
\(x^2=\left(\sqrt{21+6\sqrt{6}}-\sqrt{21-6\sqrt{6}}\right)^2\)
\(x^2=21+6\sqrt{6}+21-6\sqrt{6}-2\sqrt{441-216}\)
\(x^2=42-2\sqrt{225}\)
\(x^2=42-30=12\)
\(x=2\sqrt{3}\)
nếu có sai bn thông cảm nha
cách khác nhé:
\(\sqrt{21+6\sqrt{6}}-\sqrt{21-6\sqrt{6}}\)
\(=\sqrt{21+2.3\sqrt{2}.\sqrt{3}}-\sqrt{21-2.3\sqrt{2}.\sqrt{3}}\)
\(=\sqrt{18+2.\sqrt{18}.\sqrt{3}+3}-\sqrt{18-2.\sqrt{18}.\sqrt{3}+3}\)
\(=\sqrt{\left(\sqrt{18}+\sqrt{3}\right)^2}-\sqrt{\left(\sqrt{18}-\sqrt{3}\right)^2}\)
\(=\left(\sqrt{18}+\sqrt{3}\right)-\left(\sqrt{18}-\sqrt{3}\right)\)
\(=2\sqrt{3}\)
p/s: mk đã phân tích kĩ ra cho bn rồi đó
Tính \(A=\sqrt{21+6\sqrt{6}}+\sqrt{21-6\sqrt{6}}\)
\(A=\sqrt{\left(3\sqrt{2}\right)^2+2.3\sqrt{2}.\sqrt{3}+\left(\sqrt{3}\right)^2}+\sqrt{\left(3\sqrt{2}\right)^2-2.3.\sqrt{2}.\sqrt{3}+\left(\sqrt{3}\right)^2}\)
\(A=\sqrt{\left(3\sqrt{2}+\sqrt{3}\right)^2}+\sqrt{\left(3\sqrt{2}-\sqrt{3}\right)^2}\)
\(A=3\sqrt{2}+\sqrt{3}+3\sqrt{2}-\sqrt{3}=6\sqrt{2}\)
Tính tổng sau:
D=6/15*18+6/18*21+6/21*24+...+6/87*90
\(D=\frac{6}{15\times18}+\frac{6}{18\times21}+\frac{6}{21\times24}+...+\frac{6}{87\times90}\)
\(=2\times\left(\frac{3}{15\times18}+\frac{3}{18\times21}+...+\frac{3}{87\times90}\right)\)
\(=2\times\left(\frac{18-15}{15\times18}+\frac{21-18}{18\times21}+...+\frac{90-87}{87\times90}\right)\)
\(=2\times\left(\frac{1}{15}-\frac{1}{18}+\frac{1}{18}-\frac{1}{21}+...+\frac{1}{87}-\frac{1}{90}\right)\)
\(=2\times\left(\frac{1}{15}-\frac{1}{90}\right)=\frac{1}{9}\)
6/16 : 3/6 < x < 20/21 : 4/21. Tìm x
\(\Leftrightarrow\dfrac{6}{16}\cdot2< x< \dfrac{20}{21}\cdot\dfrac{21}{4}=5\)
=>3/4<x<5
hay \(x\in\left\{1;2;3;4\right\}\)
5/6 : 7 :20/21 + 5/6 : 7 : 20/21
kết quả bằng \(\frac{1}{4}\)
\(\sqrt[3]{27+6\sqrt{21}}+\sqrt[3]{27-6\sqrt{21}}\)
Lời giải:
Đặt \(\sqrt[3]{27+6\sqrt{21}}=a; \sqrt[3]{27-6\sqrt{21}}=b\) thì ta cần tính tổng $A=a+b$.
Ta có:
$a^3+b^3=54$
\(ab=\sqrt[3]{(27+6\sqrt{21})(27-6\sqrt{21})}=-3\)
$A^3=(a+b)^3=a^3+b^3=3ab(a+b)=54+3(-3)A$
$\Leftrightarrow A^3=54-9A$
$\Leftrightarrow A^3+9A-54=0$
$\Leftrightarrow A^2(A-3)+3A(A-3)+18(A-3)=0$
$\Leftrightarrow (A^2+3A+18)(A-3)=0$
$\Leftrightarrow A-3=0$ (do $A^2+3A+18>0$)
$\Leftrightarrow A=3$
Tính:
\(\sqrt{21+6\sqrt{6}}-\sqrt{21-6\sqrt{6}}\)
Cách khác :
\(\sqrt{21+6\sqrt{6}}-\sqrt{21-6\sqrt{6}}=\dfrac{\sqrt{36+2.6.\sqrt{6}+6}-\sqrt{36-2.6.\sqrt{6}+6}}{\sqrt{2}}=\dfrac{6+\sqrt{6}-6+\sqrt{6}}{\sqrt{2}}=\dfrac{2\sqrt{6}}{\sqrt{2}}=2\sqrt{3}\)
Đặt: \(A=\sqrt{21+6\sqrt{6}}-\sqrt{21-6\sqrt{6}}\)
\(A^2=21+6\sqrt{6}-2\sqrt{\left(21+6\sqrt{6}\right)\left(21-6\sqrt{6}\right)}+21-6\sqrt{6}\)
\(A^2=42-2\sqrt{441-216}\)
\(A^2=42-2\sqrt{225}\)
\(A^2=42-30=12\)
\(\Rightarrow A=\sqrt{12}=2\sqrt{3}\)