Hướng dẫn Giải Bài 34 (trang 19, SGK Toán 9, Tập 1)

Hướng dẫn giải Bài 34 (Trang 19 SGK Toán 9, Tập 1)

Bài 34 (Trang 19 SGK Toán 9, Tập 1): Rút gọn các biểu thức sau: <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">a</mi><mo>)</mo><mo> </mo><msup><mi>ab</mi><mn>2</mn></msup><mo>.</mo><msqrt><mfrac><mn>3</mn><mrow><msup><mi mathvariant="normal">a</mi><mn>2</mn></msup><msup><mi mathvariant="normal">b</mi><mn>4</mn></msup></mrow></mfrac></msqrt><mo> </mo><mi>với</mi><mo> </mo><mi mathvariant="normal">a</mi><mo> </mo><mo><</mo><mo> </mo><mn>0</mn><mo>,</mo><mo> </mo><mi mathvariant="normal">b</mi><mo>≠</mo><mn>0</mn><mo>;</mo></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">b</mi><mo>)</mo><mo> </mo><msqrt><mfrac><mrow><mn>27</mn><msup><mfenced><mrow><mi mathvariant="normal">a</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mn>2</mn></msup></mrow><mn>48</mn></mfrac></msqrt><mo> </mo><mi>với</mi><mo> </mo><mi mathvariant="normal">a</mi><mo> </mo><mo>></mo><mo> </mo><mn>3</mn><mo>;</mo></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">c</mi><mo>)</mo><mo> </mo><msqrt><mfrac><mrow><mn>9</mn><mo> </mo><mo>+</mo><mo> </mo><mn>12</mn><mi mathvariant="normal">a</mi><mo> </mo><mo>+</mo><mo> </mo><mn>4</mn><msup><mi mathvariant="normal">a</mi><mn>2</mn></msup></mrow><msup><mi mathvariant="normal">b</mi><mn>2</mn></msup></mfrac></msqrt><mo> </mo><mi>với</mi><mo> </mo><mi mathvariant="normal">a</mi><mo> </mo><mo>≥</mo><mo> </mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo> </mo><mi>và</mi><mo> </mo><mi mathvariant="normal">b</mi><mo> </mo><mo><</mo><mo> </mo><mn>0</mn><mo>;</mo></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">d</mi><mo>)</mo><mo> </mo><mfenced><mrow><mi mathvariant="normal">a</mi><mo> </mo><mo>-</mo><mo> </mo><mi mathvariant="normal">b</mi></mrow></mfenced><mo>.</mo><msqrt><mfrac><mi>ab</mi><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><mo> </mo><mi>với</mi><mo> </mo><mi mathvariant="normal">a</mi><mo> </mo><mo><</mo><mo> </mo><mi mathvariant="normal">b</mi><mo> </mo><mo><</mo><mo> </mo><mn>0</mn><mo>.</mo></math>   Hướng dẫn giải: <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">a</mi><mo>)</mo><mo> </mo><msup><mi>ab</mi><mn>2</mn></msup><mo>.</mo><msqrt><mfrac><mn>3</mn><mrow><msup><mi mathvariant="normal">a</mi><mn>2</mn></msup><msup><mi mathvariant="normal">b</mi><mn>4</mn></msup></mrow></mfrac></msqrt><mo> </mo><mi>với</mi><mo> </mo><mi mathvariant="normal">a</mi><mo> </mo><mo><</mo><mo> </mo><mn>0</mn><mo>,</mo><mo> </mo><mi mathvariant="normal">b</mi><mo>≠</mo><mn>0</mn></math> Ta có: <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>ab</mi><mn>2</mn></msup><mo>.</mo><msqrt><mfrac><mn>3</mn><mrow><msup><mi mathvariant="normal">a</mi><mn>2</mn></msup><msup><mi mathvariant="normal">b</mi><mn>4</mn></msup></mrow></mfrac></msqrt><mo> </mo><mo> </mo><mo>=</mo><mo> </mo><mi>a</mi><msup><mi>b</mi><mn>2</mn></msup><mo>.</mo><mfrac><msqrt><mn>3</mn></msqrt><msqrt><msup><mi>a</mi><mn>2</mn></msup><msup><mi>b</mi><mn>4</mn></msup></msqrt></mfrac><mo> </mo><mo>=</mo><mo> </mo><mi>a</mi><msup><mi>b</mi><mn>2</mn></msup><mo>.</mo><mfrac><msqrt><mn>3</mn></msqrt><msqrt><msup><mi>a</mi><mn>2</mn></msup><mo>.</mo><msqrt><msup><mi>b</mi><mn>4</mn></msup></msqrt></msqrt></mfrac><mspace linebreak="newline"/><mo> </mo><mo>=</mo><mo> </mo><mi>a</mi><msup><mi>b</mi><mn>2</mn></msup><mo>.</mo><mfrac><msqrt><mn>3</mn></msqrt><mrow><mfenced open="|" close="|"><mi>a</mi></mfenced><mo>.</mo><mfenced open="|" close="|"><msup><mi>b</mi><mn>2</mn></msup></mfenced></mrow></mfrac><mo> </mo><mo>=</mo><mo> </mo><mi>a</mi><msup><mi>b</mi><mn>2</mn></msup><mo>.</mo><mfrac><msqrt><mn>3</mn></msqrt><mrow><mo>-</mo><mi>a</mi><msup><mi>b</mi><mn>2</mn></msup></mrow></mfrac><mo> </mo><mo>=</mo><mo> </mo><mo>-</mo><msqrt><mn>3</mn></msqrt><mo> </mo><mo>(</mo><mi>V</mi><mi>ì</mi><mo> </mo><mi>a</mi><mo> </mo><mo><</mo><mo> </mo><mn>0</mn><mo>,</mo><mo> </mo><mi>b</mi><mo>≠</mo><mn>0</mn><mo> </mo><mi>n</mi><mi>ê</mi><mi>n</mi><mo> </mo><msup><mi>b</mi><mn>2</mn></msup><mo> </mo><mo>></mo><mo> </mo><mn>0</mn><mo> </mo><mo>)</mo><mo> </mo><mo>⇒</mo><mfenced open="|" close="|"><msup><mi>b</mi><mn>2</mn></msup></mfenced><mo> </mo><mo>=</mo><mo> </mo><msup><mi>b</mi><mn>2</mn></msup><mo>)</mo></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">b</mi><mo>)</mo><mo> </mo><msqrt><mfrac><mrow><mn>27</mn><msup><mfenced><mrow><mi mathvariant="normal">a</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mn>2</mn></msup></mrow><mn>48</mn></mfrac></msqrt><mo> </mo><mi>với</mi><mo> </mo><mi mathvariant="normal">a</mi><mo> </mo><mo>></mo><mo> </mo><mn>3</mn></math> Ta có: <math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mfrac><mrow><mn>27</mn><msup><mfenced><mrow><mi mathvariant="normal">a</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mn>2</mn></msup></mrow><mn>48</mn></mfrac></msqrt><mo> </mo><mo>=</mo><mo> </mo><msqrt><mfrac><mn>27</mn><mn>48</mn></mfrac><mo>.</mo><msup><mfenced><mrow><mi mathvariant="normal">a</mi><mo> </mo><mo>-</mo><mo> </mo><mn>3</mn></mrow></mfenced><mn>2</mn></msup></msqrt><mo> </mo><mo>=</mo><mo> </mo><msqrt><mfrac><mn>27</mn><mn>48</mn></mfrac></msqrt><mo>.</mo><msqrt><msup><mfenced><mrow><mi mathvariant="normal">a</mi><mo> </mo><mo>-</mo><mo> </mo><mn>3</mn></mrow></mfenced><mn>2</mn></msup></msqrt><mspace linebreak="newline"/><mo> </mo><mo>=</mo><mo> </mo><msqrt><mfrac><mrow><mn>9</mn><mo>.</mo><mn>3</mn></mrow><mrow><mn>16</mn><mo>.</mo><mn>3</mn></mrow></mfrac></msqrt><mo>.</mo><msqrt><msup><mfenced><mrow><mi mathvariant="normal">a</mi><mo> </mo><mo>-</mo><mo> </mo><mn>3</mn></mrow></mfenced><mn>2</mn></msup></msqrt><mo> </mo><mo>=</mo><mo> </mo><msqrt><mfrac><mn>9</mn><mn>16</mn></mfrac></msqrt><mo>.</mo><msqrt><msup><mfenced><mrow><mi mathvariant="normal">a</mi><mo> </mo><mo> </mo><mo>-</mo><mn>3</mn></mrow></mfenced><mn>2</mn></msup></msqrt><mspace linebreak="newline"/><mo> </mo><mo>=</mo><mo> </mo><mfrac><mn>3</mn><mn>4</mn></mfrac><mo>.</mo><msqrt><msup><mfenced><mrow><mi mathvariant="normal">a</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mn>2</mn></msup></msqrt><mo> </mo><mo>=</mo><mo> </mo><mfrac><mn>3</mn><mn>4</mn></mfrac><mo>.</mo><mfenced open="|" close="|"><mrow><mi mathvariant="normal">a</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mo> </mo><mo>=</mo><mo> </mo><mfrac><mn>3</mn><mn>4</mn></mfrac><mfenced><mrow><mi mathvariant="normal">a</mi><mo> </mo><mo> </mo><mo>-</mo><mo> </mo><mn>3</mn></mrow></mfenced><mspace linebreak="newline"/><mo mathvariant="italic">(</mo><mi>V</mi><mi>ì</mi><mo mathvariant="italic"> </mo><mi>a</mi><mo mathvariant="italic"> </mo><mo mathvariant="italic">></mo><mo mathvariant="italic"> </mo><mn mathvariant="italic">3</mn><mo mathvariant="italic"> </mo><mi>n</mi><mi>ê</mi><mi>n</mi><mo mathvariant="italic"> </mo><mi>a</mi><mo mathvariant="italic"> </mo><mo mathvariant="italic">-</mo><mo mathvariant="italic"> </mo><mn mathvariant="italic">3</mn><mo mathvariant="italic"> </mo><mo mathvariant="italic">></mo><mo mathvariant="italic"> </mo><mn mathvariant="italic">0</mn><mo mathvariant="italic"> </mo><mo mathvariant="italic">⇒</mo><mrow><mo mathvariant="italic">|</mo><mi mathvariant="italic">a</mi><mo mathvariant="italic"> </mo><mo mathvariant="italic">-</mo><mo mathvariant="italic"> </mo><mn mathvariant="italic">3</mn><mo mathvariant="italic">|</mo></mrow><mo mathvariant="italic"> </mo><mo mathvariant="italic">=</mo><mo mathvariant="italic"> </mo><mi>a</mi><mo mathvariant="italic"> </mo><mo mathvariant="italic">-</mo><mo mathvariant="italic"> </mo><mn mathvariant="italic">3</mn><mo mathvariant="italic">)</mo></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">c</mi><mo>)</mo><mo> </mo><msqrt><mfrac><mrow><mn>9</mn><mo> </mo><mo>+</mo><mo> </mo><mn>12</mn><mi mathvariant="normal">a</mi><mo> </mo><mo>+</mo><mo> </mo><mn>4</mn><msup><mi mathvariant="normal">a</mi><mn>2</mn></msup></mrow><msup><mi mathvariant="normal">b</mi><mn>2</mn></msup></mfrac></msqrt><mo> </mo><mi>với</mi><mo> </mo><mi mathvariant="normal">a</mi><mo> </mo><mo>≥</mo><mo> </mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo> </mo><mi>và</mi><mo> </mo><mi mathvariant="normal">b</mi><mo> </mo><mo><</mo><mo> </mo><mn>0</mn><mo>;</mo></math> Ta có: <math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mfrac><mrow><mn>9</mn><mo> </mo><mo>+</mo><mo> </mo><mn>12</mn><mi mathvariant="normal">a</mi><mo> </mo><mo>+</mo><mo> </mo><mn>4</mn><msup><mi mathvariant="normal">a</mi><mn>2</mn></msup></mrow><msup><mi mathvariant="normal">b</mi><mn>2</mn></msup></mfrac></msqrt><mo> </mo><mo>=</mo><mo> </mo><msqrt><mfrac><mrow><msup><mn>3</mn><mn>2</mn></msup><mo> </mo><mo>+</mo><mo> </mo><mn>2</mn><mo>.</mo><mn>3</mn><mo>.</mo><mn>2</mn><mi>a</mi><mo> </mo><mo>+</mo><mo> </mo><msup><mn>2</mn><mn>2</mn></msup><mo>.</mo><msup><mi>a</mi><mn>2</mn></msup></mrow><msup><mi>b</mi><mn>2</mn></msup></mfrac></msqrt><mspace linebreak="newline"/><mo> </mo><mo>=</mo><mo> </mo><msqrt><mfrac><mrow><msup><mn>3</mn><mn>2</mn></msup><mo> </mo><mo>+</mo><mo> </mo><mn>2</mn><mo>.</mo><mn>3</mn><mo>.</mo><mn>2</mn><mi>a</mi><mo> </mo><mo>+</mo><mo> </mo><msup><mfenced><mrow><mn>2</mn><mi>a</mi></mrow></mfenced><mn>2</mn></msup></mrow><msup><mi>b</mi><mn>2</mn></msup></mfrac></msqrt><mo> </mo><mo>=</mo><mo> </mo><msqrt><mfrac><msup><mfenced><mrow><mn>3</mn><mo> </mo><mo>+</mo><mo> </mo><mn>2</mn><mi>a</mi></mrow></mfenced><mn>2</mn></msup><msup><mi>b</mi><mn>2</mn></msup></mfrac></msqrt><mo> </mo><mo>=</mo><mo> </mo><mfrac><mfenced open="|" close="|"><mrow><mn>3</mn><mo> </mo><mo>+</mo><mo> </mo><mn>2</mn><mi>a</mi></mrow></mfenced><mfenced open="|" close="|"><mi>b</mi></mfenced></mfrac></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Vì</mi><mo> </mo><mi mathvariant="normal">a</mi><mo> </mo><mo>≥</mo><mo> </mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo> </mo><mo>⇒</mo><mo> </mo><mi mathvariant="normal">a</mi><mo> </mo><mo>+</mo><mo> </mo><mn>1</mn><mo>,</mo><mn>5</mn><mo> </mo><mo>></mo><mo> </mo><mn>0</mn><mspace linebreak="newline"/><mo>⇔</mo><mo> </mo><mn>2</mn><mo>(</mo><mi mathvariant="normal">a</mi><mo> </mo><mo>+</mo><mo> </mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo> </mo><mo>></mo><mo> </mo><mn>0</mn><mspace linebreak="newline"/><mo>⇔</mo><mn>2</mn><mi mathvariant="normal">a</mi><mo> </mo><mo> </mo><mo>+</mo><mn>3</mn><mo> </mo><mo>></mo><mo> </mo><mn>0</mn><mspace linebreak="newline"/><mo>⇔</mo><mn>3</mn><mo> </mo><mo>+</mo><mo> </mo><mn>2</mn><mi mathvariant="normal">a</mi><mo> </mo><mo>></mo><mo> </mo><mn>0</mn><mspace linebreak="newline"/><mo>⇔</mo><mfenced open="|" close="|"><mrow><mn>3</mn><mo> </mo><mo>+</mo><mo> </mo><mn>2</mn><mi mathvariant="normal">a</mi></mrow></mfenced><mo> </mo><mo>=</mo><mo> </mo><mn>3</mn><mo> </mo><mo>+</mo><mo> </mo><mn>2</mn><mi mathvariant="normal">a</mi></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Vì</mi><mo> </mo><mi mathvariant="normal">b</mi><mo> </mo><mo><</mo><mo> </mo><mn>0</mn><mo> </mo><mo>⇒</mo><mo> </mo><mfenced open="|" close="|"><mi mathvariant="normal">b</mi></mfenced><mo> </mo><mo>=</mo><mo> </mo><mo>-</mo><mi mathvariant="normal">b</mi></math> Do vậy, biểu thức <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mfenced open="|" close="|"><mrow><mn>3</mn><mo> </mo><mo>+</mo><mo> </mo><mn>2</mn><mi mathvariant="normal">a</mi></mrow></mfenced><mfenced open="|" close="|"><mi mathvariant="normal">b</mi></mfenced></mfrac><mo> </mo><mo>=</mo><mo> </mo><mfrac><mrow><mn>3</mn><mo> </mo><mo>+</mo><mo> </mo><mn>2</mn><mi mathvariant="normal">a</mi></mrow><mrow><mo>-</mo><mi mathvariant="normal">b</mi></mrow></mfrac><mo> </mo><mo>=</mo><mo>-</mo><mfrac><mrow><mn>3</mn><mo> </mo><mo>+</mo><mo> </mo><mn>2</mn><mi mathvariant="normal">a</mi></mrow><mi mathvariant="normal">b</mi></mfrac></math>. Vậy <math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mfrac><mrow><mn>9</mn><mo> </mo><mo>+</mo><mo> </mo><mn>12</mn><mi mathvariant="normal">a</mi><mo> </mo><mo>+</mo><mo> </mo><mn>4</mn><msup><mi mathvariant="normal">a</mi><mn>2</mn></msup></mrow><msup><mi mathvariant="normal">b</mi><mn>2</mn></msup></mfrac></msqrt><mo> </mo><mo>=</mo><mo> </mo><mo>-</mo><mfrac><mrow><mn>3</mn><mo> </mo><mo>+</mo><mo> </mo><mn>2</mn><mi mathvariant="normal">a</mi></mrow><mi mathvariant="normal">b</mi></mfrac></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">d</mi><mo>)</mo><mo> </mo><mfenced><mrow><mi mathvariant="normal">a</mi><mo> </mo><mo>-</mo><mo> </mo><mi mathvariant="normal">b</mi></mrow></mfenced><mo>.</mo><msqrt><mfrac><mi>ab</mi><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><mo> </mo><mi>với</mi><mo> </mo><mi mathvariant="normal">a</mi><mo> </mo><mo><</mo><mo> </mo><mi mathvariant="normal">b</mi><mo> </mo><mo><</mo><mo> </mo><mn>0</mn><mo>.</mo></math> Ta có: <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi mathvariant="normal">a</mi><mo> </mo><mo>-</mo><mo> </mo><mi mathvariant="normal">b</mi></mrow></mfenced><mo>.</mo><msqrt><mfrac><mi>ab</mi><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><mo> </mo><mo> </mo><mo>=</mo><mo> </mo><mfenced><mrow><mi>a</mi><mo> </mo><mo>-</mo><mo> </mo><mi>b</mi></mrow></mfenced><mo>.</mo><mfrac><msqrt><mi>a</mi><mi>b</mi></msqrt><msqrt><msup><mfenced><mrow><mi>a</mi><mo>-</mo><mi>b</mi></mrow></mfenced><mn>2</mn></msup></msqrt></mfrac><mo> </mo><mo>=</mo><mo> </mo><mfenced><mrow><mi>a</mi><mo>-</mo><mi>b</mi></mrow></mfenced><mfrac><msqrt><mi>a</mi><mi>b</mi></msqrt><mfenced open="|" close="|"><mrow><mi>a</mi><mo>-</mo><mi>b</mi></mrow></mfenced></mfrac><mo> </mo><mo>=</mo><mo> </mo><mfenced><mrow><mi>a</mi><mo>-</mo><mi>b</mi></mrow></mfenced><mfrac><msqrt><mi>a</mi><mi>b</mi></msqrt><mrow><mo>-</mo><mfenced><mrow><mi>a</mi><mo>-</mo><mi>b</mi></mrow></mfenced></mrow></mfrac><mo> </mo><mo>=</mo><mo> </mo><mo>-</mo><msqrt><mi>a</mi><mi>b</mi></msqrt></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>V</mi><mi>ì</mi><mo> </mo><mi>a</mi><mo> </mo><mo><</mo><mo> </mo><mi>b</mi><mo> </mo><mo><</mo><mo> </mo><mn>0</mn><mo> </mo><mi>n</mi><mi>ê</mi><mi>n</mi><mo> </mo><mi>a</mi><mo> </mo><mo>-</mo><mo> </mo><mi>b</mi><mo> </mo><mo><</mo><mo> </mo><mn>0</mn><mo> </mo><mo>⇒</mo><mo> </mo><mfenced open="|" close="|"><mrow><mi>a</mi><mo> </mo><mo>-</mo><mo> </mo><mi>b</mi></mrow></mfenced><mo> </mo><mo>=</mo><mo> </mo><mo>-</mo><mfenced><mrow><mi>a</mi><mo> </mo><mo>-</mo><mo> </mo><mi>b</mi></mrow></mfenced><mo> </mo><mi>v</mi><mi>à</mi><mo> </mo><mi>a</mi><mo>.</mo><mi>b</mi><mo> </mo><mo>></mo><mo> </mo><mn>0</mn><mo>)</mo></math>