- LG a
- LG b
- LG c
- LG d
Đơn giản biểu thức:
LG a
\[{{\sqrt a - \sqrt b } \over {\root 4 \of a - \root 4 \of b }} - {{\sqrt a + \root 4 \of {ab} } \over {\root 4 \of a + \root 4 \of b }}\]
Lời giải chi tiết:
\[{{\sqrt a - \sqrt b } \over {\root 4 \of a - \root 4 \of b }} - {{\sqrt a + \root 4 \of {ab} } \over {\root 4 \of a + \root 4 \of b }}\]
\[= {{\left[ {\root 4 \of a + \root 4 \of b } \right]\left[ {\root 4 \of a - \root 4 \of b } \right]} \over {\root 4 \of a - \root 4 \of b }} - {{\root 4 \of a \left[ {\root 4 \of a + \root 4 \of b } \right]} \over {\root 4 \of a + \root 4 \of b }}\]
\[ = \root 4 \of a + \root 4 \of b - \root 4 \of a = \root 4 \of b \]
LG b
\[{{a - b} \over {\root 3 \of a - \root 3 \of b }} - {{a + b} \over {\root 3 \of a + \root 3 \of b }}\]
Lời giải chi tiết:
\[{{a - b} \over {\root 3 \of a - \root 3 \of b }} - {{a + b} \over {\root 3 \of a + \root 3 \of b }} \]
\[= {{{{\left[ {\root 3 \of a } \right]}^3} - {{\left[ {\root 3 \of b } \right]}^3}} \over {\root 3 \of a - \root 3 \of b }} - {{{{\left[ {\root 3 \of a } \right]}^3} + {{\left[ {\root 3 \of b } \right]}^3}} \over {\root 3 \of a + \root 3 \of b }}\]
\[ = \frac{{\left[ {\sqrt[3]{a} - \sqrt[3]{b}} \right]\left[ {\sqrt[3]{{{a^2}}} + \sqrt[3]{{ab}} + \sqrt[3]{{{b^2}}}} \right]}}{{\left[ {\sqrt[3]{a} - \sqrt[3]{b}} \right]}}\] \[ - \frac{{\left[ {\sqrt[3]{a} + \sqrt[3]{b}} \right]\left[ {\sqrt[3]{{{a^2}}} - \sqrt[3]{{ab}} + \sqrt[3]{{{b^2}}}} \right]}}{{\left[ {\sqrt[3]{a} + \sqrt[3]{b}} \right]}}\]
\[ = [\root 3 \of {{a^2}} + \root 3 \of {ab} + \root 3 \of {{b^2}}] \] \[ - \left[ {\root 3 \of {{a^2}} - \root 3 \of {ab} + \root 3 \of {{b^2}} } \right] \]
\[= 2\root 3 \of {ab} \]
LG c
\[\left[ {{{a + b} \over {\root 3 \of a + \root 3 \of b }} - \root 3 \of {ab} } \right]:{\left[ {\root 3 \of a - \root 3 \of b } \right]^2};\]
Lời giải chi tiết:
\[\left[ {{{a + b} \over {\root 3 \of a + \root 3 \of b }} - \root 3 \of {ab} } \right]:{\left[ {\root 3 \of a - \root 3 \of b } \right]^2}\]
\[ = [\frac{{\left[ {\sqrt[3]{a} + \sqrt[3]{b}} \right]\left[ {\sqrt[3]{{{a^2}}} - \sqrt[3]{{ab}} + \sqrt[3]{{{b^2}}}} \right]}}{{\left[ {\sqrt[3]{a} + \sqrt[3]{b}} \right]}}- \root 3 \of {ab}]:\] \[:{\left[ {\sqrt[3]{a} - \sqrt[3]{b}} \right]^2}\]
\[= \left[ {\root 3 \of {{a^2}} - \root 3 \of {ab} + \root 3 \of {{b^2}} - \root 3 \of {ab} } \right]:\] \[:{\left[ {\root 3 \of a - \root 3 \of b } \right]^2}\]
\[ = \left[ {\root 3 \of {{a^2}} - 2\root 3 \of {ab} + \root 3 \of {{b^2}} } \right]:{\left[ {\root 3 \of a - \root 3 \of b } \right]^2} \]
\[= {\left[ {\root 3 \of a - \root 3 \of b } \right]^2}:{\left[ {\root 3 \of a - \root 3 \of b } \right]^2} = 1\]
LG d
\[{{a - 1} \over {{a^{{3 \over 4}}} + {a^{{1 \over 2}}}}}.{{\sqrt a + \root 4 \of a } \over {\sqrt a + 1}}.{a^{{1 \over 4}}} + 1.\]
Lời giải chi tiết:
\[{{a - 1} \over {{a^{{3 \over 4}}} + {a^{{1 \over 2}}}}}.{{\sqrt a + \root 4 \of a } \over {\sqrt a + 1}}.{a^{{1 \over 4}}} + 1. \]
\[= {{\left[ {\sqrt a + 1} \right]\left[ {\sqrt a - 1} \right]} \over {\sqrt[4]{{{a^3}}} + \sqrt a }}.{{\root 4 \of a \left[ {\root 4 \of a + 1} \right]} \over {\left[ {\sqrt a + 1} \right]}}.\root 4 \of a + 1\]
\[ = \frac{{\left[ {\sqrt a + 1} \right]\left[ {\sqrt[4]{a} + 1} \right]\left[ {\sqrt[4]{a} - 1} \right]}}{{\sqrt a \left[ {\sqrt[4]{a} + 1} \right]}}.\frac{{\sqrt[4]{a}\left[ {\sqrt[4]{a} + 1} \right]}}{{\sqrt a + 1}}.\sqrt[4]{a} + 1\]
\[ = \frac{{\left[ {\sqrt[4]{a} + 1} \right]\left[ {\sqrt[4]{a} - 1} \right].{{\left[ {\sqrt[4]{a}} \right]}^2}}}{{\sqrt a }} + 1\]
\[ = \sqrt a - 1 + 1 = \sqrt a \].
Cách khác: