Rút Gọn
a)\(S=\sqrt{\frac{36a^2b^6c^8}{4}}\) với a < 0; b < 0
b)\(S=\sqrt{\frac{1}{abc}\left(\sqrt{\frac{abc^2}{4}+\sqrt{\frac{ab^5c^3}{9}}}\right)}\) với a > 0 ; b > 0 ; c > 0
QV
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Rút gọn P
P=\(\left(\frac{a+\sqrt{a^2-b^2}}{a-\sqrt{a^2-b^2}}-\frac{a-\sqrt{a^2-b^2}}{a+\sqrt{a^2-b^2}}\right):\frac{4\sqrt{a^4-a^2b^2}}{b^2}\)
với |a|>|b|>0
LL
30 tháng 6 2021 lúc 16:30
* Rút gọn:
a.\(\sqrt{\left(\sqrt{7}-4\right)^2+\sqrt{7}}\)
b.\(\sqrt{81a}-\sqrt{144a}+\sqrt{36a}\) với a≥0
`a)sqrt{(sqrt7-4)^2}+sqrt7`
`=|sqrt7-4|+sqrt7`
`=4-sqrt7+sqrt7=4`
`b)\sqrt{81a}-sqrt{144a}+sqrt{36a}(a>=0)`
`=9sqrta-12sqrta+6sqrta=3sqrta`
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a) Ta có: \(\sqrt{\left(\sqrt{7}-4\right)^2}+\sqrt{7}\)
\(=4-\sqrt{7}+\sqrt{7}\)
=4
b) Ta có: \(\sqrt{81a}-\sqrt{144a}+\sqrt{36a}\)
\(=9\sqrt{a}-12\sqrt{a}+6\sqrt{a}\)
\(=3\sqrt{a}\)
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Rút gọn:
a) \(\sqrt{\frac{2a^2b^4}{50}}\)
b) \(\frac{\sqrt{2ab^2}}{\sqrt{162}}\) với a \(\ge\) 0
a) \(\sqrt{\frac{2a^2b^4}{50}}=\sqrt{\frac{a^2b^4}{25}}=\frac{\sqrt{a^2b^4}}{\sqrt{25}}=\frac{ab^2}{5}\)
b) \(\frac{\sqrt{2ab^2}}{\sqrt{162}}=\sqrt{\frac{2ab^2}{162}}=\sqrt{\frac{ab^2}{81}}=\frac{\sqrt{ab^2}}{\sqrt{81}}=\frac{b\sqrt{a}}{9}\)
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\(B=\frac{\sqrt{a+4\sqrt{a-4}}+\sqrt{a-4\sqrt{a-4}}}{\sqrt{1-\frac{8}{a}+\frac{16}{a^{^2}}}}\)RÚT GỌN B với a>8
\(=\frac{\sqrt{a+4\sqrt{a-4}}+\sqrt{a-4\sqrt{a-4}}}{\sqrt{1-\frac{8}{a}+\frac{16}{a^2}}}\)
\(=\frac{\sqrt{\left(\sqrt{a-4}+2\right)^2}+\sqrt{\left(\sqrt{a-4}\right)-2}}{\sqrt{\left(1-\frac{4}{a}\right)^2}}\)
\(=\frac{\sqrt{a-4}+2+\sqrt{a-4}-2}{1-\frac{4}{a}}\)
\(=\frac{2a}{\sqrt{a-4}}\)
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Rút gọn biểu thức
a. \(\frac{2b\sqrt{x^2-1}}{x-\sqrt{x^2-1}}\) với \(x=\frac{1}{2}\left(\sqrt{\frac{a}{b}}+\sqrt{\frac{b}{a}}\right)\)và và a>0 b>0
rút gọn và tính giá trị biểu thức:
\(\frac{a-b}{\sqrt{a\left(a+2b\right)+b}}:\sqrt{\frac{\left(a-b\right)^2}{a\left(a+b\right)}}với\)
\(a>0;b>0;\frac{a}{b}=\frac{9}{7}\)
Rút gọn:
\(S=\sqrt{\frac{1}{a^2+b^2}+\frac{1}{\left(a+b\right)^2}+\sqrt{\frac{1}{a^4+b^4}+\frac{1}{\left(a^2+b^2\right)^2}}}\\ \)
với a, b>0
rút gọn P: P=\(\left(\frac{a+\sqrt{a^2-b^2}}{a-\sqrt{a^2}-b^2}-\frac{a-\sqrt{a^2-b^2}}{a+\sqrt{a^2-b^2}}\right):\frac{4\sqrt{a^4-a^2b^2}}{b^2}\)
\(P=\left(\frac{a+\sqrt{a^2-b^2}}{a-\sqrt{a^2-b^2}}-\frac{a-\sqrt{a^2-b^2}}{a+\sqrt{a^2-b^2}}\right):\frac{4\sqrt{a^4-a^2b^2}}{b^2}\)
\(=\left[\frac{\left(a+\sqrt{a^2-b^2}\right)\left(a+\sqrt{a^2-b^2}\right)-\left(a-\sqrt{a^2-b^2}\right)\left(a-\sqrt{a^2-b^2}\right)}{\left(a-\sqrt{a^2-b^2}\right)\left(a+\sqrt{a^2-b^2}\right)}\right]:\frac{4\sqrt{a^2\left(a^2-b^2\right)}}{b^2}\)
\(=\left[\frac{\left(a+\sqrt{a^2-b^2}\right)^2-\left(a-\sqrt{a^2-b^2}\right)}{a^2-\left(a^2-b^2\right)}\right]:\frac{4a\sqrt{a^2-b^2}}{b^2}\)
\(=\frac{\left(a+\sqrt{a^2-b^2}+a-\sqrt{a^2-b^2}\right)\left(a+\sqrt{a^2-b^2}-a+\sqrt{a^2-b^2}\right)}{b^2}\cdot\frac{b^2}{4a\sqrt{a^2-b^2}}\)
\(=\frac{2a\cdot2\sqrt{a^2-b^2}}{b^2}\cdot\frac{b^2}{4a\sqrt{a^2-b^2}}\)
\(=1\)
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Rút gọn
\(B=\frac{2}{x^2-y^2}\sqrt{\frac{9\left(x^2+2xy+y^2\right)}{4}}\) với x > -y
\(C=\sqrt{\frac{2a}{3}}.\sqrt{\frac{3a}{8}}\) với a >hoặc= 0
\(\frac{1}{a-b}\sqrt{a^4\left(a-b\right)^2}\) với a > 0
\(B=\frac{2}{x^2-y^2}\cdot\sqrt{\frac{9\left(x^2+2xy+y^2\right)}{4}}=\frac{2}{\left(x-y\right)\left(x+y\right)}\cdot\sqrt{\frac{9\left(x+y\right)^2}{4}}\)
\(=\frac{2}{\left(x-y\right)\left(x+y\right)}\cdot\frac{\sqrt{9\left(x+y\right)^2}}{\sqrt{4}}=\frac{2}{\left(x-y\right)\left(x+y\right)}\cdot\frac{3\left(x+y\right)}{2}\)(vì x > -y <=> x + y > 0)
\(=\frac{3}{x-y}\)
\(C=\sqrt{\frac{2a}{3}}.\sqrt{\frac{3a}{8}}=\sqrt{\frac{2a}{3}\cdot\frac{3a}{8}}=\sqrt{\frac{6a^2}{24}}=\sqrt{\frac{a^2}{4}}=\frac{a}{2}\)(vì a > = 0)
\(D=\frac{1}{a-b}\cdot\sqrt{a^4\left(a-b\right)^2}=\frac{1}{a-b}\cdot a^2\left(a-b\right)=a^2\)(a > b > 0)
câu cuối điều kiện là a>b
\(\frac{1}{a-b}\sqrt{a^4\left(a-b\right)^2}=\frac{a^2\left|a-b\right|}{a-b}=\frac{a^2\left(a-b\right)}{a-b}=a^2\) (vì a>b)