Hướng dẫn giải Bài 65 (Trang 34 SGK Toán 9, Tập 1)

Bài 65 (Trang 33 SGK Toán 9, Tập 1): Rút gọn rồi so sánh giá trị của M với 1, biết: <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">M</mi><mo> </mo><mo>=</mo><mo> </mo><mfenced><mrow><mfrac><mn>1</mn><mrow><mi mathvariant="normal">a</mi><mo> </mo><mo>-</mo><msqrt><mi mathvariant="normal">a</mi></msqrt></mrow></mfrac><mo>+</mo><mfrac><mn>1</mn><mrow><msqrt><mi mathvariant="normal">a</mi></msqrt><mo>-</mo><mn>1</mn></mrow></mfrac></mrow></mfenced><mo>:</mo><mfrac><mrow><msqrt><mi mathvariant="normal">a</mi></msqrt><mo>+</mo><mn>1</mn></mrow><mrow><mi mathvariant="normal">a</mi><mo>-</mo><mn>2</mn><msqrt><mi mathvariant="normal">a</mi></msqrt><mo>+</mo><mn>1</mn></mrow></mfrac><mo> </mo><mi>v</mi><mi>ớ</mi><mi>i</mi><mo> </mo><mi>a</mi><mo> </mo><mo>></mo><mo> </mo><mn>0</mn><mo> </mo><mi>v</mi><mi>à</mi><mo> </mo><mi>a</mi><mo> </mo><mo>≠</mo><mo> </mo><mn>1</mn><mo>.</mo></math>   Hướng dẫn giải: <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">M</mi><mo> </mo><mo>=</mo><mo> </mo><mfenced><mrow><mfrac><mn>1</mn><mrow><mi mathvariant="normal">a</mi><mo> </mo><mo>-</mo><msqrt><mi mathvariant="normal">a</mi></msqrt></mrow></mfrac><mo>+</mo><mfrac><mn>1</mn><mrow><msqrt><mi mathvariant="normal">a</mi></msqrt><mo>-</mo><mn>1</mn></mrow></mfrac></mrow></mfenced><mo>:</mo><mfrac><mrow><msqrt><mi mathvariant="normal">a</mi></msqrt><mo>+</mo><mn>1</mn></mrow><mrow><mi mathvariant="normal">a</mi><mo>-</mo><mn>2</mn><msqrt><mi mathvariant="normal">a</mi></msqrt><mo>+</mo><mn>1</mn></mrow></mfrac><mo>=</mo><mo> </mo><mfrac><mrow><msqrt><mi mathvariant="normal">a</mi></msqrt><mo>-</mo><mn>1</mn><mo>+</mo><mi mathvariant="normal">a</mi><mo>-</mo><msqrt><mi mathvariant="normal">a</mi></msqrt></mrow><mrow><mfenced><mrow><mi mathvariant="normal">a</mi><mo>-</mo><msqrt><mi mathvariant="normal">a</mi></msqrt></mrow></mfenced><mfenced><mrow><msqrt><mi mathvariant="normal">a</mi></msqrt><mo>-</mo><mn>1</mn></mrow></mfenced></mrow></mfrac><mo>:</mo><mfrac><mrow><msqrt><mi mathvariant="normal">a</mi></msqrt><mo>-</mo><mn>1</mn></mrow><msup><mfenced><mrow><msqrt><mi mathvariant="normal">a</mi></msqrt><mo>-</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup></mfrac><mspace linebreak="newline"/><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo>=</mo><mo> </mo><mfrac><mrow><mi>a</mi><mo>-</mo><mn>1</mn></mrow><mrow><mfenced><mrow><mi>a</mi><mo>-</mo><msqrt><mi>a</mi></msqrt></mrow></mfenced><mfenced><mrow><msqrt><mi>a</mi></msqrt><mo>-</mo><mn>1</mn></mrow></mfenced></mrow></mfrac><mo>.</mo><mfrac><msup><mfenced><mrow><msqrt><mi>a</mi></msqrt><mo>-</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup><mrow><msqrt><mi>a</mi></msqrt><mo>+</mo><mn>1</mn></mrow></mfrac><mo>=</mo><mfrac><mrow><msup><mfenced><msqrt><mi>a</mi></msqrt></mfenced><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow><mrow><mfenced><mrow><mi>a</mi><mo>-</mo><msqrt><mi>a</mi></msqrt></mrow></mfenced><mfenced><mrow><msqrt><mi>a</mi></msqrt><mo>-</mo><mn>1</mn></mrow></mfenced></mrow></mfrac><mo>.</mo><mfrac><msup><mfenced><msqrt><mi>a</mi><mo>-</mo><mn>1</mn></msqrt></mfenced><mn>2</mn></msup><mrow><msqrt><mi>a</mi></msqrt><mo>+</mo><mn>1</mn></mrow></mfrac><mspace linebreak="newline"/><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo>=</mo><mo> </mo><mfrac><mrow><mfenced><mrow><mi>a</mi><mo> </mo><mo>-</mo><mo> </mo><msqrt><mi>a</mi></msqrt></mrow></mfenced><mfenced><mrow><msqrt><mi>a</mi></msqrt><mo> </mo><mo>+</mo><mo> </mo><mn>1</mn></mrow></mfenced><msup><mfenced><mrow><msqrt><mi>a</mi></msqrt><mo>-</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup></mrow><mrow><mfenced><mrow><mi>a</mi><mo> </mo><mo>-</mo><mo> </mo><msqrt><mi>a</mi></msqrt></mrow></mfenced><mfenced><mrow><msqrt><mi>a</mi></msqrt><mo> </mo><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><msqrt><mi>a</mi></msqrt><mo> </mo><mo>+</mo><mo> </mo><mn>1</mn></mrow></mfenced></mrow></mfrac><mo> </mo><mo>=</mo><mo> </mo><mfrac><msup><mfenced><mrow><msqrt><mi>a</mi></msqrt><mo> </mo><mo>-</mo><mo> </mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup><mrow><msup><mfenced><msqrt><mi>a</mi></msqrt></mfenced><mn>2</mn></msup><mo> </mo><mo>-</mo><mo> </mo><msqrt><mi>a</mi></msqrt></mrow></mfrac><mspace linebreak="newline"/><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo>=</mo><mo> </mo><mfrac><msup><mfenced><mrow><msqrt><mi>a</mi></msqrt><mo> </mo><mo>-</mo><mo> </mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup><mrow><msqrt><mi>a</mi></msqrt><mfenced><mrow><msqrt><mi>a</mi></msqrt><mo> </mo><mo>-</mo><mo> </mo><mn>1</mn></mrow></mfenced></mrow></mfrac><mo> </mo><mo>=</mo><mo> </mo><mfrac><mrow><msqrt><mi>a</mi></msqrt><mo> </mo><mo>-</mo><mo> </mo><mn>1</mn></mrow><msqrt><mi>a</mi></msqrt></mfrac><mo> </mo><mo>=</mo><mo> </mo><mn>1</mn><mo> </mo><mo>-</mo><mo> </mo><mfrac><mn>1</mn><msqrt><mi>a</mi></msqrt></mfrac><mo> </mo><mo><</mo><mo> </mo><mn>1</mn></math> Vậy M < 1.