Hướng dẫn Giải Bài 64 (trang 33, SGK Toán 9, Tập 1)

Hướng dẫn giải Bài 64 (Trang 33 SGK Toán 9, Tập 1)

Bài 64 (Trang 33 SGK Toán 9, Tập 1): Chứng minh các đẳng thức sau: <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">a</mi><mo>)</mo><mo> </mo><mfenced><mrow><mfrac><mrow><mn>1</mn><mo>-</mo><mi mathvariant="normal">a</mi><msqrt><mi mathvariant="normal">a</mi></msqrt></mrow><mrow><mn>1</mn><mo>-</mo><msqrt><mi mathvariant="normal">a</mi></msqrt></mrow></mfrac><mo>+</mo><msqrt><mi mathvariant="normal">a</mi></msqrt></mrow></mfenced><msup><mfenced><mfrac><mrow><mn>1</mn><mo>-</mo><msqrt><mi mathvariant="normal">a</mi></msqrt></mrow><mrow><mn>1</mn><mo>-</mo><mi mathvariant="normal">a</mi></mrow></mfrac></mfenced><mn>2</mn></msup><mo> </mo><mo>=</mo><mo> </mo><mn>1</mn><mo> </mo><mi>v</mi><mi>ớ</mi><mi>i</mi><mo mathvariant="italic"> </mo><mi>a</mi><mo mathvariant="italic"> </mo><mo mathvariant="italic">></mo><mo mathvariant="italic"> </mo><mn mathvariant="italic">0</mn><mo mathvariant="italic"> </mo><mi>v</mi><mi>à</mi><mo mathvariant="italic"> </mo><mi>a</mi><mo mathvariant="italic"> </mo><mo mathvariant="italic">≠</mo><mn mathvariant="italic">1</mn><mo mathvariant="italic">;</mo></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">b</mi><mo>)</mo><mo> </mo><mfrac><mrow><mi mathvariant="normal">a</mi><mo> </mo><mo>+</mo><mo> </mo><mi mathvariant="normal">b</mi></mrow><msup><mi mathvariant="normal">b</mi><mn>2</mn></msup></mfrac><mo>.</mo><mo> </mo><msqrt><mfrac><mrow><msup><mi mathvariant="normal">a</mi><mn>2</mn></msup><msup><mi mathvariant="normal">b</mi><mn>4</mn></msup></mrow><mrow><msup><mi mathvariant="normal">a</mi><mrow><mn>2</mn><mo> </mo></mrow></msup><mo>+</mo><mo> </mo><mn>2</mn><mi>ab</mi><mo> </mo><mo>+</mo><mo> </mo><msup><mi mathvariant="normal">b</mi><mn>2</mn></msup></mrow></mfrac></msqrt><mo> </mo><mo>=</mo><mfenced open="|" close="|"><mi mathvariant="normal">a</mi></mfenced><mo> </mo><mi>v</mi><mi>ớ</mi><mi>i</mi><mo mathvariant="italic"> </mo><mi>a</mi><mo mathvariant="italic"> </mo><mo mathvariant="italic">+</mo><mo mathvariant="italic"> </mo><mi>b</mi><mo mathvariant="italic"> </mo><mo mathvariant="italic">></mo><mo mathvariant="italic"> </mo><mn mathvariant="italic">0</mn><mo mathvariant="italic"> </mo><mi>v</mi><mi>à</mi><mo mathvariant="italic"> </mo><mi>b</mi><mo mathvariant="italic">≠</mo><mn mathvariant="italic">0</mn><mo mathvariant="italic">.</mo></math>   Hướng dẫn giải: a) Thực hiện vế trái: <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>VT</mi><mo> </mo><mo>=</mo><mo> </mo><mfenced><mrow><mfrac><mrow><mn>1</mn><mo>-</mo><mi mathvariant="normal">a</mi><msqrt><mi mathvariant="normal">a</mi></msqrt></mrow><mrow><mn>1</mn><mo>-</mo><msqrt><mi mathvariant="normal">a</mi></msqrt></mrow></mfrac><mo> </mo><mo>+</mo><msqrt><mi mathvariant="normal">a</mi></msqrt></mrow></mfenced><msup><mfenced><mfrac><mrow><mn>1</mn><mo>-</mo><msqrt><mi mathvariant="normal">a</mi></msqrt></mrow><mrow><mn>1</mn><mo>-</mo><mi mathvariant="normal">a</mi></mrow></mfrac></mfenced><mn>2</mn></msup><mo>=</mo><mo> </mo><mfrac><mrow><mn>1</mn><mo>-</mo><mi mathvariant="normal">a</mi><msqrt><mi mathvariant="normal">a</mi></msqrt><mo>+</mo><msqrt><mi mathvariant="normal">a</mi></msqrt><mo>-</mo><mi mathvariant="normal">a</mi></mrow><mrow><mn>1</mn><mo>-</mo><msqrt><mi mathvariant="normal">a</mi></msqrt></mrow></mfrac><mo>.</mo><mfrac><msup><mfenced><mrow><mn>1</mn><mo>-</mo><msqrt><mi mathvariant="normal">a</mi></msqrt></mrow></mfenced><mn>2</mn></msup><msup><mfenced><mrow><mn>1</mn><mo>-</mo><mi mathvariant="normal">a</mi></mrow></mfenced><mn>2</mn></msup></mfrac><mspace linebreak="newline"/><mspace linebreak="newline"/><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo>=</mo><mo> </mo><mfenced><mrow><mn>1</mn><mo>-</mo><mi mathvariant="normal">a</mi><msqrt><mi mathvariant="normal">a</mi></msqrt><mo>+</mo><mo> </mo><msqrt><mi mathvariant="normal">a</mi></msqrt><mo> </mo><mo>-</mo><mi mathvariant="normal">a</mi></mrow></mfenced><mfrac><mrow><mn>1</mn><mo>-</mo><msqrt><mi mathvariant="normal">a</mi></msqrt></mrow><msup><mfenced><mrow><mn>1</mn><mo>-</mo><mi mathvariant="normal">a</mi></mrow></mfenced><mn>2</mn></msup></mfrac><mo> </mo><mo>=</mo><mo> </mo><mfrac><mrow><mn>1</mn><mo>-</mo><mi mathvariant="normal">a</mi><msqrt><mi mathvariant="normal">a</mi></msqrt><mo>+</mo><msqrt><mi mathvariant="normal">a</mi></msqrt><mo>-</mo><mi mathvariant="normal">a</mi><mo>-</mo><msqrt><mi mathvariant="normal">a</mi></msqrt><mo>+</mo><mi mathvariant="normal">a</mi><mo>.</mo><msqrt><msup><mi mathvariant="normal">a</mi><mn>2</mn></msup></msqrt><mo>-</mo><msup><mfenced><msqrt><mi mathvariant="normal">a</mi></msqrt></mfenced><mn>2</mn></msup><mo>+</mo><mi mathvariant="normal">a</mi><msqrt><mi mathvariant="normal">a</mi></msqrt></mrow><msup><mfenced><mrow><mn>1</mn><mo>-</mo><mi mathvariant="normal">a</mi></mrow></mfenced><mn>2</mn></msup></mfrac><mspace linebreak="newline"/><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo>=</mo><mo> </mo><mfrac><mrow><msup><mi mathvariant="normal">a</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi mathvariant="normal">a</mi><mo>+</mo><mn>1</mn></mrow><msup><mfenced><mrow><mn>1</mn><mo>-</mo><mi mathvariant="normal">a</mi></mrow></mfenced><mn>2</mn></msup></mfrac><mo>=</mo><mfrac><msup><mfenced><mrow><mi mathvariant="normal">a</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup><msup><mfenced><mrow><mn>1</mn><mo>-</mo><mi mathvariant="normal">a</mi></mrow></mfenced><mn>2</mn></msup></mfrac><mo> </mo><mo>=</mo><mo> </mo><msup><mfenced><mfrac><mrow><mi mathvariant="normal">a</mi><mo>-</mo><mn>1</mn></mrow><mrow><mn>1</mn><mo>-</mo><mi mathvariant="normal">a</mi></mrow></mfrac></mfenced><mn>2</mn></msup><mo>=</mo><mo> </mo><mfenced><mrow><mo>-</mo><mn>1</mn></mrow></mfenced><mo> </mo><mo>=</mo><mn>1</mn><mo>=</mo><mo> </mo><mi>VP</mi><mo> </mo><mo>(</mo><mi>đ</mi><mi>ẳ</mi><mi>n</mi><mi>g</mi><mo> </mo><mi>t</mi><mi>h</mi><mi>ứ</mi><mi>c</mi><mo> </mo><mi>đ</mi><mi>ư</mi><mi>ợ</mi><mi>c</mi><mo> </mo><mi>c</mi><mi>h</mi><mi>ứ</mi><mi>n</mi><mi>g</mi><mo> </mo><mi>m</mi><mi>i</mi><mi>n</mi><mi>h</mi><mo>)</mo></math>   b) Thực hiện vế trái: <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>VT</mi><mo> </mo><mo>=</mo><mo> </mo><mfrac><mrow><mi mathvariant="normal">a</mi><mo> </mo><mo>+</mo><mo> </mo><mi mathvariant="normal">b</mi></mrow><msup><mi mathvariant="normal">b</mi><mn>2</mn></msup></mfrac><mo>.</mo><mo> </mo><msqrt><mfrac><mrow><msup><mi mathvariant="normal">a</mi><mn>2</mn></msup><msup><mi mathvariant="normal">b</mi><mn>4</mn></msup></mrow><mrow><msup><mi mathvariant="normal">a</mi><mn>2</mn></msup><mo> </mo><mo>+</mo><mo> </mo><mn>2</mn><mi>ab</mi><mo> </mo><mo>+</mo><mo> </mo><msup><mi mathvariant="normal">b</mi><mn>2</mn></msup></mrow></mfrac></msqrt><mo>=</mo><mo> </mo><mfrac><mrow><mi mathvariant="normal">a</mi><mo> </mo><mo>+</mo><mo> </mo><mi mathvariant="normal">b</mi></mrow><msup><mi mathvariant="normal">b</mi><mn>2</mn></msup></mfrac><mo>.</mo><msqrt><mfrac><msup><mfenced><msup><mi>ab</mi><mn>2</mn></msup></mfenced><mn>2</mn></msup><msup><mfenced><mrow><mi mathvariant="normal">a</mi><mo> </mo><mo>+</mo><mo> </mo><mi mathvariant="normal">b</mi></mrow></mfenced><mn>2</mn></msup></mfrac></msqrt><mspace linebreak="newline"/><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo>=</mo><mo> </mo><mfrac><mrow><mi mathvariant="normal">a</mi><mo>+</mo><mi mathvariant="normal">b</mi></mrow><msup><mi mathvariant="normal">b</mi><mn>2</mn></msup></mfrac><mo>.</mo><mfrac><mfenced open="|" close="|"><msup><mi>ab</mi><mn>2</mn></msup></mfenced><mfenced open="|" close="|"><mrow><mi mathvariant="normal">a</mi><mo>+</mo><mi mathvariant="normal">b</mi></mrow></mfenced></mfrac><mo> </mo><mo>=</mo><mo> </mo><mfrac><mrow><mi mathvariant="normal">a</mi><mo>+</mo><mi mathvariant="normal">b</mi></mrow><msup><mi mathvariant="normal">b</mi><mn>2</mn></msup></mfrac><mo>.</mo><mfrac><mrow><msup><mi mathvariant="normal">b</mi><mn>2</mn></msup><mfenced open="|" close="|"><mi mathvariant="normal">a</mi></mfenced></mrow><mrow><mi mathvariant="normal">a</mi><mo>+</mo><mi mathvariant="normal">b</mi></mrow></mfrac><mo> </mo><mo>=</mo><mfenced open="|" close="|"><mi mathvariant="normal">a</mi></mfenced><mo> </mo><mo>=</mo><mo> </mo><mi>VP</mi><mo> </mo><mo mathvariant="italic">(</mo><mi>đ</mi><mi>ẳ</mi><mi>n</mi><mi>g</mi><mo mathvariant="italic"> </mo><mi>t</mi><mi>h</mi><mi>ứ</mi><mi>c</mi><mo mathvariant="italic"> </mo><mi>đ</mi><mi>ư</mi><mi>ợ</mi><mi>c</mi><mo mathvariant="italic"> </mo><mi>c</mi><mi>h</mi><mi>ứ</mi><mi>n</mi><mi>g</mi><mo mathvariant="italic"> </mo><mi>m</mi><mi>i</mi><mi>n</mi><mi>h</mi><mo mathvariant="italic">)</mo></math>