Hướng dẫn giải Bài 75 (Trang 40 SGK Toán 9, Tập 1)

Bài 75 (Trang 41 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>2</mn><msqrt><mn>3</mn></msqrt><mo>-</mo><msqrt><mn>6</mn></msqrt></mrow><mrow><msqrt><mn>8</mn></msqrt><mo>-</mo><mn>2</mn></mrow></mfrac><mo> </mo><mo>-</mo><mo> </mo><mfrac><msqrt><mn>216</mn></msqrt><mn>3</mn></mfrac></mrow></mfenced><mo>.</mo><mfrac><mn>1</mn><msqrt><mn>6</mn></msqrt></mfrac><mo> </mo><mo>=</mo><mo> </mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>;</mo></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">b</mi><mo>)</mo><mo> </mo><mfenced><mrow><mfrac><mrow><msqrt><mn>14</mn></msqrt><mo>-</mo><msqrt><mn>7</mn></msqrt></mrow><mrow><mn>1</mn><mo>-</mo><msqrt><mn>2</mn></msqrt></mrow></mfrac><mo>+</mo><mfrac><mrow><msqrt><mn>15</mn></msqrt><mo>-</mo><msqrt><mn>5</mn></msqrt></mrow><mrow><mn>1</mn><mo>-</mo><msqrt><mn>3</mn></msqrt></mrow></mfrac></mrow></mfenced><mo>:</mo><mfrac><mn>1</mn><mrow><msqrt><mn>7</mn></msqrt><mo>-</mo><msqrt><mn>5</mn></msqrt></mrow></mfrac><mo>=</mo><mo> </mo><mo>-</mo><mn>2</mn><mo>;</mo></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">c</mi><mo>)</mo><mo> </mo><mfrac><mrow><mi mathvariant="normal">a</mi><msqrt><mi mathvariant="normal">b</mi></msqrt><mo>+</mo><mi mathvariant="normal">b</mi><msqrt><mi mathvariant="normal">a</mi></msqrt></mrow><msqrt><mi>ab</mi></msqrt></mfrac><mo>:</mo><mfrac><mn>1</mn><mrow><msqrt><mi mathvariant="normal">a</mi></msqrt><mo>-</mo><msqrt><mi mathvariant="normal">b</mi></msqrt></mrow></mfrac><mo> </mo><mo>=</mo><mo> </mo><mi mathvariant="normal">a</mi><mo> </mo><mo>-</mo><mo> </mo><mi mathvariant="normal">b</mi><mo> </mo><mi>với</mi><mo> </mo><mi mathvariant="normal">a</mi><mo>,</mo><mo> </mo><mi mathvariant="normal">b</mi><mo> </mo><mo>></mo><mo> </mo><mn>0</mn><mo> </mo><mi>và</mi><mo> </mo><mi mathvariant="normal">a</mi><mo>≠</mo><mi mathvariant="normal">b</mi><mo>;</mo></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">d</mi><mo>)</mo><mo> </mo><mfenced><mrow><mn>1</mn><mo> </mo><mo>+</mo><mo> </mo><mfrac><mrow><mi mathvariant="normal">a</mi><mo>+</mo><msqrt><mi mathvariant="normal">a</mi></msqrt></mrow><mrow><msqrt><mi mathvariant="normal">a</mi></msqrt><mo>+</mo><mn>1</mn></mrow></mfrac></mrow></mfenced><mfenced><mrow><mn>1</mn><mo> </mo><mo>-</mo><mo> </mo><mfrac><mrow><mi mathvariant="normal">a</mi><mo>-</mo><msqrt><mi mathvariant="normal">a</mi></msqrt></mrow><mrow><msqrt><mi mathvariant="normal">a</mi></msqrt><mo>-</mo><mn>1</mn></mrow></mfrac></mrow></mfenced><mo> </mo><mo>=</mo><mo> </mo><mo> </mo><mn>1</mn><mo> </mo><mo>-</mo><mo> </mo><mi mathvariant="normal">a</mi><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><mn>1</mn><mo>.</mo></math>   Hướng dẫn giải: a) <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>VT</mi><mo> </mo><mo>=</mo><mo> </mo><mo> </mo><mfenced><mrow><mfrac><mrow><mn>2</mn><msqrt><mn>3</mn></msqrt><mo>-</mo><msqrt><mn>6</mn></msqrt></mrow><mrow><msqrt><mn>8</mn></msqrt><mo>-</mo><mn>2</mn></mrow></mfrac><mo> </mo><mo>-</mo><mo> </mo><mfrac><msqrt><mn>216</mn></msqrt><mn>3</mn></mfrac></mrow></mfenced><mo>.</mo><mfrac><mn>1</mn><msqrt><mn>6</mn></msqrt></mfrac><mspace linebreak="newline"/><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo>=</mo><mo> </mo><mfenced><mrow><mfrac><mrow><msqrt><msup><mn>2</mn><mn>2</mn></msup><mo>.</mo><mn>3</mn></msqrt><mo>-</mo><msqrt><mn>6</mn></msqrt></mrow><mrow><msqrt><mn>4</mn><mo>.</mo><mn>2</mn></msqrt><mo>-</mo><mn>2</mn></mrow></mfrac><mo> </mo><mo>-</mo><mfrac><msqrt><mn>36</mn><mo>.</mo><mn>6</mn></msqrt><mn>3</mn></mfrac></mrow></mfenced><mfrac><mn>1</mn><msqrt><mn>6</mn></msqrt></mfrac><mspace linebreak="newline"/><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo>=</mo><mo> </mo><mfenced><mrow><mfrac><mrow><msqrt><mn>2</mn></msqrt><mo>.</mo><msqrt><mn>6</mn></msqrt><mo>-</mo><msqrt><mn>6</mn></msqrt></mrow><mrow><mn>2</mn><mfenced><mrow><msqrt><mn>2</mn></msqrt><mo>-</mo><mn>1</mn></mrow></mfenced></mrow></mfrac><mo> </mo><mo>-</mo><mo> </mo><mn>2</mn><msqrt><mn>6</mn></msqrt></mrow></mfenced><mfrac><mn>1</mn><msqrt><mn>6</mn></msqrt></mfrac><mspace linebreak="newline"/><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo>=</mo><mfenced><mrow><mfrac><msqrt><mn>6</mn></msqrt><mn>2</mn></mfrac><mo> </mo><mo>-</mo><mo> </mo><mn>2</mn><msqrt><mn>6</mn></msqrt></mrow></mfenced><mfrac><mn>1</mn><msqrt><mn>6</mn></msqrt></mfrac><mspace linebreak="newline"/><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo>=</mo><mfrac><mrow><mo>-</mo><mn>3</mn><msqrt><mn>6</mn></msqrt></mrow><mn>2</mn></mfrac><mo>.</mo><mfrac><mn>1</mn><msqrt><mn>6</mn></msqrt></mfrac><mo> </mo><mo>=</mo><mo> </mo><mn>1</mn><mo>,</mo><mn>5</mn><mo> </mo><mo> </mo><mo>=</mo><mo> </mo><mi>VP</mi></math> b) <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>VT</mi><mo> </mo><mo>=</mo><mo> </mo><mfenced><mrow><mfrac><mrow><msqrt><mn>14</mn></msqrt><mo>-</mo><msqrt><mn>7</mn></msqrt></mrow><mrow><mn>1</mn><mo>-</mo><msqrt><mn>2</mn></msqrt></mrow></mfrac><mo> </mo><mo>+</mo><mo> </mo><mfrac><mrow><msqrt><mn>15</mn></msqrt><mo>-</mo><msqrt><mn>5</mn></msqrt></mrow><mrow><mn>1</mn><mo>-</mo><msqrt><mn>3</mn></msqrt></mrow></mfrac></mrow></mfenced><mo>:</mo><mfrac><mn>1</mn><mrow><msqrt><mn>7</mn></msqrt><mo>-</mo><msqrt><mn>5</mn></msqrt></mrow></mfrac><mo> </mo><mo>=</mo><mo> </mo><mo>-</mo><mn>2</mn><mspace linebreak="newline"/><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo>=</mo><mo> </mo><mfenced open="[" close="]"><mrow><mfrac><mrow><msqrt><mn>7</mn></msqrt><mo>.</mo><msqrt><mn>2</mn></msqrt><mo>-</mo><msqrt><mn>7</mn></msqrt></mrow><mrow><mn>1</mn><mo> </mo><mo>-</mo><msqrt><mn>2</mn></msqrt></mrow></mfrac><mo>+</mo><mfrac><mrow><msqrt><mn>5</mn></msqrt><mo>.</mo><msqrt><mn>3</mn></msqrt><mo>-</mo><msqrt><mn>5</mn></msqrt></mrow><mrow><mn>1</mn><mo>-</mo><msqrt><mn>3</mn></msqrt></mrow></mfrac></mrow></mfenced><mo>:</mo><mfrac><mn>1</mn><mrow><msqrt><mn>7</mn></msqrt><mo>-</mo><msqrt><mn>5</mn></msqrt></mrow></mfrac><mspace linebreak="newline"/><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo>=</mo><mfenced open="[" close="]"><mrow><mfrac><mrow><msqrt><mn>7</mn></msqrt><mfenced><mrow><msqrt><mn>2</mn></msqrt><mo>-</mo><mn>1</mn></mrow></mfenced></mrow><mrow><mn>1</mn><mo>-</mo><msqrt><mn>2</mn></msqrt></mrow></mfrac><mo>+</mo><mfrac><mrow><msqrt><mn>5</mn></msqrt><mfenced><mrow><msqrt><mn>3</mn></msqrt><mo>-</mo><mn>1</mn></mrow></mfenced></mrow><mrow><mn>1</mn><mo>-</mo><msqrt><mn>3</mn></msqrt></mrow></mfrac></mrow></mfenced><mo>.</mo><mfenced><mrow><msqrt><mn>7</mn></msqrt><mo>-</mo><msqrt><mn>5</mn></msqrt></mrow></mfenced><mspace linebreak="newline"/><mspace linebreak="newline"/><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo>=</mo><mfenced><mrow><mo>-</mo><msqrt><mn>7</mn></msqrt><mo>-</mo><msqrt><mn>5</mn></msqrt></mrow></mfenced><mfenced><mrow><msqrt><mn>7</mn></msqrt><mo>-</mo><msqrt><mn>5</mn></msqrt></mrow></mfenced><mspace linebreak="newline"/><mspace linebreak="newline"/><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo>=</mo><mo>-</mo><mfenced><mrow><msqrt><mn>7</mn></msqrt><mo>+</mo><msqrt><mn>5</mn></msqrt></mrow></mfenced><mfenced><mrow><msqrt><mn>7</mn></msqrt><mo>-</mo><msqrt><mn>5</mn></msqrt></mrow></mfenced><mspace linebreak="newline"/><mspace linebreak="newline"/><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo>=</mo><mo>-</mo><mfenced><mrow><mn>7</mn><mo>-</mo><mn>5</mn></mrow></mfenced><mo> </mo><mo>=</mo><mo> </mo><mo>-</mo><mn>2</mn><mo> </mo><mo>=</mo><mo> </mo><mi>VP</mi></math> c) <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><msqrt><mi mathvariant="normal">b</mi></msqrt><mo>+</mo><mi mathvariant="normal">b</mi><msqrt><mi mathvariant="normal">a</mi></msqrt></mrow><msqrt><mi>ab</mi></msqrt></mfrac><mo>:</mo><mfrac><mn>1</mn><mrow><msqrt><mi mathvariant="normal">a</mi></msqrt><mo>-</mo><msqrt><mi mathvariant="normal">b</mi></msqrt></mrow></mfrac><mspace linebreak="newline"/><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo>=</mo><mo> </mo><mfrac><mrow><msqrt><msup><mi>a</mi><mn>2</mn></msup></msqrt><mo>.</mo><msqrt><mi>b</mi></msqrt><mo>+</mo><msqrt><msup><mi>b</mi><mn>2</mn></msup><mo>.</mo><msqrt><mi>a</mi></msqrt></msqrt></mrow><msqrt><mi>a</mi><mi>b</mi></msqrt></mfrac><mo>:</mo><mfrac><mn>1</mn><mrow><msqrt><mi>a</mi></msqrt><mo>-</mo><msqrt><mi>b</mi></msqrt></mrow></mfrac><mspace linebreak="newline"/><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo>=</mo><mfrac><mrow><msqrt><mi>a</mi><mi>b</mi></msqrt><mfenced><mrow><msqrt><mi>a</mi></msqrt><mo>+</mo><msqrt><mi>b</mi></msqrt></mrow></mfenced></mrow><msqrt><mi>a</mi><mi>b</mi></msqrt></mfrac><mo>.</mo><mfenced><mrow><msqrt><mi>a</mi></msqrt><mo>-</mo><msqrt><mi>b</mi></msqrt></mrow></mfenced><mspace linebreak="newline"/><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo>=</mo><mfenced><mrow><msqrt><mi>a</mi></msqrt><mo>+</mo><msqrt><mi>b</mi></msqrt></mrow></mfenced><mfenced><mrow><msqrt><mi>a</mi></msqrt><mo>-</mo><msqrt><mi>b</mi></msqrt></mrow></mfenced><mo> </mo><mo>=</mo><mo> </mo><mi>a</mi><mo> </mo><mo>-</mo><mo> </mo><mi>b</mi><mo> </mo><mo>=</mo><mo> </mo><mi>V</mi><mi>P</mi></math>             d) <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>VT</mi><mo> </mo><mo>=</mo><mo> </mo><mfenced><mrow><mn>1</mn><mo> </mo><mo>+</mo><mo> </mo><mfrac><mrow><mi mathvariant="normal">a</mi><mo>+</mo><msqrt><mi mathvariant="normal">a</mi></msqrt></mrow><mrow><msqrt><mi mathvariant="normal">a</mi></msqrt><mo>+</mo><mn>1</mn></mrow></mfrac></mrow></mfenced><mfenced><mrow><mn>1</mn><mo> </mo><mo>-</mo><mo> </mo><mfrac><mrow><mi mathvariant="normal">a</mi><mo>-</mo><msqrt><mi mathvariant="normal">a</mi></msqrt></mrow><mrow><msqrt><mi mathvariant="normal">a</mi></msqrt><mo>-</mo><mn>1</mn></mrow></mfrac></mrow></mfenced><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><mo>+</mo><mo> </mo><mfrac><mrow><msqrt><msup><mi mathvariant="normal">a</mi><mn>2</mn></msup></msqrt><mo>+</mo><msqrt><mi mathvariant="normal">a</mi></msqrt></mrow><mrow><msqrt><mi mathvariant="normal">a</mi></msqrt><mo>+</mo><mn>1</mn></mrow></mfrac></mrow></mfenced><mfenced><mrow><mn>1</mn><mo> </mo><mo>-</mo><mo> </mo><mfrac><mrow><msqrt><msup><mi mathvariant="normal">a</mi><mn>2</mn></msup></msqrt><mo>-</mo><msqrt><mi mathvariant="normal">a</mi></msqrt></mrow><mrow><msqrt><mi mathvariant="normal">a</mi></msqrt><mo>-</mo><mn>1</mn></mrow></mfrac></mrow></mfenced><mspace linebreak="newline"/><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo>=</mo><mo> </mo><mfenced open="[" close="]"><mrow><mn>1</mn><mo> </mo><mo>+</mo><mo> </mo><mfrac><mrow><msqrt><mi mathvariant="normal">a</mi></msqrt><mfenced><mrow><msqrt><mi mathvariant="normal">a</mi></msqrt><mo>+</mo><mn>1</mn></mrow></mfenced></mrow><mrow><msqrt><mi mathvariant="normal">a</mi></msqrt><mo>+</mo><mn>1</mn></mrow></mfrac></mrow></mfenced><mfenced open="[" close="]"><mrow><mn>1</mn><mo> </mo><mo>-</mo><mo> </mo><mfrac><mrow><msqrt><mi mathvariant="normal">a</mi></msqrt><mfenced><mrow><msqrt><mi mathvariant="normal">a</mi></msqrt><mo>-</mo><mn>1</mn></mrow></mfenced></mrow><mrow><msqrt><mi mathvariant="normal">a</mi></msqrt><mo>-</mo><mn>1</mn></mrow></mfrac></mrow></mfenced><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><mo>+</mo><mo> </mo><msqrt><mi mathvariant="normal">a</mi></msqrt></mrow></mfenced><mfenced><mrow><mn>1</mn><mo> </mo><mo>-</mo><mo> </mo><msqrt><mi mathvariant="normal">a</mi></msqrt></mrow></mfenced><mo> </mo><mo>=</mo><mo> </mo><mn>1</mn><mo> </mo><mo>-</mo><mo> </mo><mfenced><msqrt><msup><mi mathvariant="normal">a</mi><mn>2</mn></msup></msqrt></mfenced><mo> </mo><mo>=</mo><mo> </mo><mn>1</mn><mo>-</mo><mn>1</mn><mo> </mo><mo>=</mo><mi>VP</mi><mo> </mo></math>