Toán Lớp 9: rút gọn biểu thức A=√(5+√21)+√(5-√21)-2√(4-√7)
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Toán Lớp 9: rút gọn biểu thức A=√(5+√21)+√(5-√21)-2√(4-√7)
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A = \sqrt {5 + \sqrt {21} } + \sqrt {5 – \sqrt {21} } – 2\sqrt {4 – \sqrt 7 } \\
\Leftrightarrow \sqrt 2 .A\\
= \sqrt {10 + 2\sqrt {21} } + \sqrt {10 – 2\sqrt {21} } – 2\sqrt {8 – 2\sqrt 7 } \\
= \sqrt {{{\left( {\sqrt 7 + \sqrt 3 } \right)}^2}} + \sqrt {{{\left( {\sqrt 7 – \sqrt 3 } \right)}^2}} – 2\sqrt {{{\left( {\sqrt 7 – 1} \right)}^2}} \\
= \sqrt 7 + \sqrt 3 + \sqrt 7 – \sqrt 3 – 2.\left( {\sqrt 7 – 1} \right)\\
= 2\sqrt 7 – 2\sqrt 7 + 2\\
= 2\\
\Leftrightarrow A = \dfrac{2}{{\sqrt 2 }} = \sqrt 2
\end{array}$