Đề bài

Chứng minh: a) \(\left( {\sqrt {2025}  - \sqrt {2024} } \right)\left( {\sqrt {2025}  + \sqrt {2024} } \right) = 1\) b) \(\left( {\sqrt[3]{3} - 1} \right)\left[ {{{\left( {\sqrt[3]{3}} \right)}^2} + \sqrt[3]{3} + 1} \right] = 2\)

c) \({\left( {\sqrt 3  - 2} \right)^2}{\left( {\sqrt 3  + 2} \right)^2} = 1\)

Lời giải chi tiết

a) \(VT = \left( {\sqrt {2025}  - \sqrt {2024} } \right)\left( {\sqrt {2025}  + \sqrt {2024} } \right)\) \(\begin{array}{l} = {\left( {\sqrt {2025} } \right)^2} - {\left( {\sqrt {2024} } \right)^2}\\ = 2025 - 2024\end{array}\) \( = 1 = VP.\)(đpcm) b) \(VT = \left( {\sqrt[3]{3} - 1} \right)\left[ {{{\left( {\sqrt[3]{3}} \right)}^2} + \sqrt[3]{3} + 1} \right]\) \( = {\left( {\sqrt[3]{3}} \right)^3} - 1\) \( = 3 - 1 = 2 = VP\) (đpcm). c) \(VT = {\left( {\sqrt 3  - 2} \right)^2}{\left( {\sqrt 3  + 2} \right)^2}\) \( = {\left[ {\left( {\sqrt 3  - 2} \right)\left( {\sqrt 3  + 2} \right)} \right]^2} = {\left[ {{{\left( {\sqrt 3 } \right)}^2} - 4} \right]^2}\) \( = {\left( {3 - 4} \right)^2} = {\left( { - 1} \right)^2} = 1 = VP\) (đpcm).