Hướng dẫn giải Bài 38 (Trang 17 SGK Toán Đại số 8, Tập 1)

Chứng minh các đẳng thức sau: <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>)</mo><mo> </mo><msup><mrow><mo>(</mo><mi>a</mi><mo>−</mo><mi>b</mi><mo>)</mo></mrow><mn>3</mn></msup><mo>=</mo><mo>−</mo><msup><mrow><mo>(</mo><mi>b</mi><mo>−</mo><mi>a</mi><mo>)</mo></mrow><mn>3</mn></msup><mo>;</mo></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>)</mo><mo> </mo><msup><mrow><mo>(</mo><mo>−</mo><mi>a</mi><mo>−</mo><mi>b</mi><mo>)</mo></mrow><mn>2</mn></msup><mo>=</mo><msup><mrow><mo>(</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo>)</mo></mrow><mn>2</mn></msup><mo>;</mo></math> Giải: <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>)</mo><mo> </mo><msup><mrow><mo>(</mo><mi>a</mi><mo>−</mo><mi>b</mi><mo>)</mo></mrow><mn>3</mn></msup><mo>=</mo><mo>−</mo><msup><mrow><mo>(</mo><mi>b</mi><mo>−</mo><mi>a</mi><mo>)</mo></mrow><mn>3</mn></msup></math> Biến đổi vế phải thành vế trái: <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>−</mo><msup><mrow><mo>(</mo><mi>b</mi><mo>−</mo><mi>a</mi><mo>)</mo></mrow><mn>3</mn></msup><mo> </mo><mo>=</mo><mo>−</mo><mo>(</mo><msup><mi>b</mi><mn>3</mn></msup><mo>−</mo><mn>3</mn><msup><mi>b</mi><mn>2</mn></msup><mi>a</mi><mo>+</mo><mn>3</mn><mi>b</mi><msup><mi>a</mi><mn>2</mn></msup><mo>−</mo><msup><mi>a</mi><mn>3</mn></msup><mo>)</mo><mspace linebreak="newline"/><mo>=</mo><mo>−</mo><msup><mi>b</mi><mn>3</mn></msup><mo>+</mo><mn>3</mn><msup><mi>b</mi><mn>2</mn></msup><mi>a</mi><mo>−</mo><mn>3</mn><mi>b</mi><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>a</mi><mn>3</mn></msup><mspace linebreak="newline"/><mo>=</mo><msup><mi>a</mi><mn>3</mn></msup><mo>−</mo><mn>3</mn><msup><mi>a</mi><mn>2</mn></msup><mi>b</mi><mo>+</mo><mn>3</mn><mi>a</mi><msup><mi>b</mi><mn>2</mn></msup><mo>−</mo><msup><mi>b</mi><mn>3</mn></msup><mo>=</mo><msup><mrow><mo>(</mo><mi>a</mi><mo>−</mo><mi>b</mi><mo>)</mo></mrow><mn>3</mn></msup></math> Sử dụng tính chất hai số đối nhau: <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mo>(</mo><mi>a</mi><mo> </mo><mo>–</mo><mo> </mo><mi>b</mi><mo>)</mo></mrow><mn>3</mn></msup><mo>=</mo><mo> </mo><msup><mrow><mo>[</mo><mo>(</mo><mo>-</mo><mn>1</mn><mo>)</mo><mo>(</mo><mi>b</mi><mo> </mo><mo>–</mo><mo> </mo><mi>a</mi><mo>)</mo><mo>]</mo></mrow><mn>3</mn></msup><mo> </mo><mo>=</mo><mo> </mo><mo>(</mo><mo>-</mo><mn>1</mn><msup><mo>)</mo><mn>3</mn></msup><msup><mrow><mo>(</mo><mi>b</mi><mo> </mo><mo>–</mo><mo> </mo><mi>a</mi><mo>)</mo></mrow><mn>3</mn></msup><mo> </mo><mo>=</mo><mo> </mo><mo>-</mo><msup><mn>1</mn><mn>3</mn></msup><mo> </mo><mo>.</mo><mo> </mo><msup><mrow><mo>(</mo><mi>b</mi><mo> </mo><mo>–</mo><mo> </mo><mi>a</mi><mo>)</mo></mrow><mn>3</mn></msup><mo>=</mo><mo>-</mo><msup><mrow><mo>(</mo><mi>b</mi><mo> </mo><mo>–</mo><mo> </mo><mi>a</mi><mo>)</mo></mrow><mn>3</mn></msup></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>)</mo><mo> </mo><msup><mrow><mo>(</mo><mo>−</mo><mi>a</mi><mo>−</mo><mi>b</mi><mo>)</mo></mrow><mn>2</mn></msup><mo>=</mo><msup><mrow><mo>(</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo>)</mo></mrow><mn>2</mn></msup></math> Biến đổi vế trái thành vế phải: <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mo>(</mo><mo>−</mo><mi>a</mi><mo>−</mo><mi>b</mi><mo>)</mo></mrow><mn>2</mn></msup><mo> </mo><mspace linebreak="newline"/><mo>=</mo><msup><mrow><mo>[</mo><mo>(</mo><mo>−</mo><mi>a</mi><mo>)</mo><mo>+</mo><mo>(</mo><mo>−</mo><mi>b</mi><mo>)</mo><mo>]</mo></mrow><mn>2</mn></msup><mo> </mo><mspace linebreak="newline"/><mo>=</mo><mo>(</mo><mo>−</mo><mi>a</mi><msup><mo>)</mo><mn>2</mn></msup><mo>+</mo><mn>2</mn><mo>.</mo><mo>(</mo><mo>−</mo><mi>a</mi><mo>)</mo><mo>.</mo><mo>(</mo><mo>−</mo><mi>b</mi><mo>)</mo><mo>+</mo><msup><mrow><mo>(</mo><mo>−</mo><mi>b</mi><mo>)</mo></mrow><mn>2</mn></msup><mo> </mo><mspace linebreak="newline"/><mo>=</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>a</mi><mi>b</mi><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup><mspace linebreak="newline"/><mo>=</mo><msup><mrow><mo>(</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo>)</mo></mrow><mn>2</mn></msup></math> Sử dụng tính chất hai số đối nhau: <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mo>(</mo><mo>-</mo><mi>a</mi><mo> </mo><mo>–</mo><mo> </mo><mi>b</mi><mo>)</mo></mrow><mn>2</mn></msup><mo> </mo><mo>=</mo><mo> </mo><msup><mrow><mo>[</mo><mo>(</mo><mo>-</mo><mn>1</mn><mo>)</mo><mo> </mo><mo>.</mo><mo> </mo><mo>(</mo><mi>a</mi><mo> </mo><mo>+</mo><mo> </mo><mi>b</mi><mo>)</mo><mo>]</mo></mrow><mn>2</mn></msup><mo> </mo><mo>=</mo><mo> </mo><mo>(</mo><mo>-</mo><mn>1</mn><msup><mo>)</mo><mn>2</mn></msup><mo> </mo><mo>.</mo><mo> </mo><msup><mrow><mo>(</mo><mi>a</mi><mo> </mo><mo>+</mo><mo> </mo><mi>b</mi><mo>)</mo></mrow><mn>2</mn></msup><mo> </mo><mo>=</mo><mo> </mo><mn>1</mn><mo> </mo><mo>.</mo><mo> </mo><msup><mrow><mo>(</mo><mi>a</mi><mo> </mo><mo>+</mo><mo> </mo><mi>b</mi><mo>)</mo></mrow><mn>2</mn></msup><mo>=</mo><msup><mrow><mo>(</mo><mi>a</mi><mo> </mo><mo>+</mo><mo> </mo><mi>b</mi><mo>)</mo></mrow><mn>2</mn></msup></math>