RÚT GỌN
B = 3 + 3^2 + 3^3 + 3^4 + ... + 3^50
C = 4 + 4^2 + 4^3 + 4^4 + ...+ 4^2018
Rút gọn
B=\(\sqrt{7+4\sqrt{3}}\)-\(2\sqrt{3}\)
\(B=\sqrt{7+4\sqrt{3}}-2\sqrt{3}\)
\(=2+\sqrt{3}-2\sqrt{3}\)
\(=2-\sqrt{3}\)
Cho biểu thức
A= \(\dfrac{2}{\sqrt{x}+4}-\dfrac{3}{\sqrt{x}-4}-\dfrac{2\sqrt{x}+16}{16-x}\:\:\:\left(x\ge0,x\ne16\right)\)
a) Rút gọn
b) Tìm giá trị A khi x = \(4-2\sqrt{3}\)
Lời giải:
a.
\(A=\frac{2(\sqrt{x}-4)-3(\sqrt{x}+4)}{(\sqrt{x}-4)(\sqrt{x}+4)}+\frac{2\sqrt{x}+16}{(\sqrt{x}-4)(\sqrt{x}+4)}=\frac{-\sqrt{x}-20}{(\sqrt{x}-4)(\sqrt{x}+4)}+\frac{2\sqrt{x}+16}{(\sqrt{x}-4)(\sqrt{x}+4)}\\ =\frac{\sqrt{x}-4}{(\sqrt{x}-4)(\sqrt{x}+4)}=\frac{1}{\sqrt{x}+4}\)
b. Khi $x=4-2\sqrt{3}=(\sqrt{3}-1)^2\Rightarrow \sqrt{x}=\sqrt{3}-1$
$A=\frac{1}{\sqrt{3}-1+4}=\frac{1}{\sqrt{3}+3}$
Tính tổng sau:
a) 1 - 2 + 3 - 4 + ... + 99 - 100
b) 2 - 4 + 6 - 8 + ... + 48 - 50
c) 1 + 2 - 3 + 4 - ... - 99 + 100
a: từ 1 đến 100 sẽ có \(\dfrac{100-1}{1}+1=100-1+1=100\left(số\right)\)
=>Sẽ có \(\dfrac{100}{2}=50\) cặp số
1-2+3-4+...+99-100
=(1-2)+(3-4)+...+(99-100)
=(-1)+(-1)+...+(-1)
=-1*50=-50
b: Sửa đề: \(2-4+6-8+...+46-48+50\)
Từ 2 đến 48 sẽ có \(\dfrac{48-2}{2}+1=24-1+1=24\left(số\right)\)
=>Sẽ có \(\dfrac{24}{2}=12\left(cặp\right)\)
\(2-4+6-8+...+46-48+50\)
\(=\left(2-4\right)+\left(6-8\right)+...+\left(46-48\right)+50\)
\(=\left(-2\right)+\left(-2\right)+...+\left(-2\right)+50\)
\(=50-2\cdot24=50-48=2\)
c: Đặt A=\(1+2-3+4+...+97+98-99+100\)
\(=\left(1+2-3+4\right)+\left(5+6-7+8\right)+...+\left(97+98-99+100\right)\)
\(=4+12+...+196\)
Từ 4 đến 196 sẽ có \(\dfrac{196-4}{8}+1=\dfrac{192}{8}+1=25\left(số\right)\)
Tổng của dãy A là: \(\left(196+4\right)\cdot\dfrac{25}{2}=\dfrac{25}{2}\cdot200=100\cdot25=2500\)
a.1-2+3-4+......+99-100
b.2-4+6-8+......-48+50
c.1+2-3-4+.......+97+98-99-100
a: 1-2+3-4+...+99-100
=(1-2)+(3-4)+...+(99-100)
=(-1)+(-1)+...+(-1)
=-1*50=-50
c: 1+2-3-4+....+97+98-99-100
=(1+2-3-4)+(5+6-7-8)+...+(97+98-99-100)
=(-4)+(-4)+...+(-4)
=(-4)*25=-100
rút gọn
a)3 √5 + √20-2 √5
b)2 √2+ √8+ √50
c) 4√3+ √ 27 -√45 +2 √5
d) √ 75+ √ 48- √300
e)( √28- √12- √7) √7 +2 √21
f)( √99- √18- √11) √11+3 √22
g)3 √45-5 √125x +7 √20x+28(x>=0)
a: \(=3\sqrt{5}+2\sqrt{5}-2\sqrt{5}=3\sqrt{5}\)
b: \(=2\sqrt{2}+2\sqrt{2}+5\sqrt{2}=9\sqrt{2}\)
c: \(=4\sqrt{3}+3\sqrt{3}-3\sqrt{5}+2\sqrt{5}=7\sqrt{3}-\sqrt{5}\)
d: \(=5\sqrt{3}+4\sqrt{3}-10\sqrt{3}=-\sqrt{3}\)
e: \(=\left(\sqrt{7}-2\sqrt{3}\right)\cdot\sqrt{7}+2\sqrt{21}\)
=7-2*căn 21+2*căn 21
=7
f: \(=\left(2\sqrt{11}-3\sqrt{2}\right)\cdot\sqrt{11}+3\sqrt{22}\)
=22-3*căn 22+3*căn 22
=22
a) \(3\sqrt{5}+\sqrt{20}-2\sqrt{5}\)
\(=3\sqrt{5}+2\sqrt{5}-2\sqrt{5}\)
\(=3\sqrt{5}\)
b) \(2\sqrt{2}+\sqrt{8}+\sqrt{50}\)
\(=2\sqrt{2}+2\sqrt{2}+5\sqrt{2}\)
\(=9\sqrt{5}\)
c) \(4\sqrt{3}+\sqrt{27}-\sqrt{45}+2\sqrt{5}\)
\(=4\sqrt{3}+3\sqrt{3}-3\sqrt{5}+2\sqrt{5}\)
\(=7\sqrt{3}-\sqrt{5}\)
d) \(\sqrt{75}+\sqrt{48}-\sqrt{300}\)
\(=5\sqrt{3}+4\sqrt{3}-10\sqrt{3}\)
\(=-\sqrt{3}\)
e) \(\left(\sqrt{28}-\sqrt{12}-\sqrt{7}\right)\sqrt{7}+2\sqrt{21}\)
\(=\left(2\sqrt{7}-2\sqrt{3}-\sqrt{7}\right)\sqrt{7}+2\sqrt{21}\)
\(=\left(\sqrt{7}-2\sqrt{3}\right)\sqrt{7}+2\sqrt{21}\)
\(=7-2\sqrt{21}+2\sqrt{21}\)
\(=7\)
f) \(\left(\sqrt{99}-\sqrt{18}-\sqrt{11}\right)\sqrt{11}+3\sqrt{22}\)
\(=\left(3\sqrt{11}-3\sqrt{2}-\sqrt{11}\right)\sqrt{11}+3\sqrt{22}\)
\(=\left(2\sqrt{11}-3\sqrt{2}\right)\sqrt{11}+3\sqrt{22}\)
\(=22-3\sqrt{22}+3\sqrt{22}\)
\(=22\)
g) \(3\sqrt{45}-5\sqrt{125x}+7\sqrt{20x}+28\)
\(=9\sqrt{5}-25\sqrt{5x}+14\sqrt{5x}+28\)
\(=9\sqrt{5}-11\sqrt{5x}+28\)
Cho A=1/4+2/4^2+3/4^3+...............+2018/4^2018
M=\(\dfrac{\sqrt{x}}{\sqrt{x}-2}-\dfrac{4\sqrt{x}-4}{\sqrt{x}\left(\sqrt{x}-2\right)}\)
a) Rút gọn
b) Tính giá trị của M khi x= \(3+2\sqrt{2}\)
c) Tìm giá trị của x để M>0
Lời giải:
a. ĐKXĐ: $x>0; x\neq 4$
\(M=\frac{x}{\sqrt{x}(\sqrt{x}-2)}-\frac{4\sqrt{x}-4}{\sqrt{x}(\sqrt{x}-2)}=\frac{x-(4\sqrt{x}-4)}{\sqrt{x}(\sqrt{x}-2)}=\frac{x-4\sqrt{x}+4}{\sqrt{x}(\sqrt{x}-2)}=\frac{(\sqrt{x}-2)^2}{\sqrt{x}(\sqrt{x}-2)}=\frac{\sqrt{x}-2}{\sqrt{x}}\)
b.
\(x=3+2\sqrt{2}=(\sqrt{2}+1)^2\Rightarrow \sqrt{x}=\sqrt{2}+1\)
\(M=\frac{\sqrt{x}-2}{\sqrt{x}}=\frac{\sqrt{2}+1-2}{\sqrt{2}+1}=3-2\sqrt{2}\)
c.
$M>0\Leftrightarrow \frac{\sqrt{x}-2}{\sqrt{x}}>0$
$\Leftrightarrow \sqrt{x}-2>0$
$\Leftrightarrow \sqrt{x}>2\Leftrightarrow x>4$
Kết hợp đkxđ suy ra $x>4$
Cho biểu thức A = \(\dfrac{x+4}{\sqrt{x}+4}\) ; B = \(\left(\dfrac{x+3\sqrt{x}-3}{x-16}-\dfrac{1}{\sqrt{x}+4}\right):\dfrac{\sqrt{x}+1}{\sqrt{x}-4}\)
x ≥ 0, x ≠ 16
a) Rút gọn
b) Tìm giá trị nhỏ nhất của biểu thức \(\dfrac{A}{B}\)
(mink đag cần gấp)
a) Ta có: \(B=\left(\dfrac{x+3\sqrt{x}-3}{x-16}-\dfrac{1}{\sqrt{x}+4}\right):\dfrac{\sqrt{x}+1}{\sqrt{x}-4}\)
\(=\left(\dfrac{x+3\sqrt{x}-3-\sqrt{x}+4}{\left(\sqrt{x}+4\right)\left(\sqrt{x}-4\right)}\right):\dfrac{\sqrt{x}+1}{\sqrt{x}-4}\)
\(=\dfrac{x+2\sqrt{x}+1}{\left(\sqrt{x}+4\right)\left(\sqrt{x}-4\right)}\cdot\dfrac{\sqrt{x}-4}{\sqrt{x}+1}\)
\(=\dfrac{\sqrt{x}+1}{\sqrt{x}+4}\)
rút gọn phân số
2/1+3/2+4/3+......+2019/2018
2/1+3/2+4/3+.........+2019/2018
=2019 +1/2+1/3+....+1/2018
(biểu thức ko thể rút gọn đc nx)