Hướng dẫn giải Bài 1 (Trang 90 SGK Toán Giải Tích 12)

Bài 1 (Trang 90 SGK Toán Giải Tích 12): Hãy nêu các tính chất của lũy thừa với số mũ thực. Hướng dẫn trả lời: Tính chất của lũy thừa với số mũ thực: Cho a, b là những số thực dương; <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">α</mi><mo>,</mo><mo> </mo><mi mathvariant="normal">β</mi></math> là những số thực tùy ý. Khi đó ta có: <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="bold">a</mi><mi mathvariant="bold">α</mi></msup><mo mathvariant="bold">.</mo><msup><mi mathvariant="bold">a</mi><mi mathvariant="bold">β</mi></msup><mo mathvariant="bold"> </mo><mo mathvariant="bold">=</mo><mo mathvariant="bold"> </mo><msup><mi mathvariant="bold">a</mi><mrow><mi mathvariant="bold">α</mi><mo mathvariant="bold">+</mo><mi mathvariant="bold">β</mi></mrow></msup></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msup><mi mathvariant="bold">a</mi><mi mathvariant="bold">α</mi></msup><msup><mi mathvariant="bold">a</mi><mi mathvariant="bold">β</mi></msup></mfrac><mo mathvariant="bold"> </mo><mo mathvariant="bold">=</mo><mo mathvariant="bold"> </mo><msup><mi mathvariant="bold">a</mi><mrow><mi mathvariant="bold">α</mi><mo mathvariant="bold">-</mo><mi mathvariant="bold">β</mi></mrow></msup></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mstyle mathvariant="bold"><mrow><mo>(</mo><msup><mi>a</mi><mi>α</mi></msup><mo>)</mo></mrow></mstyle><mi mathvariant="bold">β</mi></msup><mo mathvariant="bold"> </mo><mo mathvariant="bold">=</mo><mo mathvariant="bold"> </mo><msup><mi mathvariant="bold">a</mi><mrow><mi mathvariant="bold">α</mi><mo mathvariant="bold">.</mo><mi mathvariant="bold">β</mi></mrow></msup></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mstyle mathvariant="bold"><mrow><mo>(</mo><mi>a</mi><mo>.</mo><mi>b</mi><mo>)</mo></mrow></mstyle><mi mathvariant="bold">α</mi></msup><mo mathvariant="bold"> </mo><mo mathvariant="bold">=</mo><mo mathvariant="bold"> </mo><msup><mi mathvariant="bold">a</mi><mi mathvariant="bold">α</mi></msup><mo mathvariant="bold">.</mo><msup><mi mathvariant="bold">b</mi><mi mathvariant="bold">β</mi></msup></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mstyle mathvariant="bold"><mrow><mo>(</mo><mfrac><mi>a</mi><mi>b</mi></mfrac><mo>)</mo></mrow></mstyle><mi mathvariant="bold">α</mi></msup><mo mathvariant="bold"> </mo><mo mathvariant="bold">=</mo><mo mathvariant="bold"> </mo><mfrac><msup><mi mathvariant="bold">a</mi><mi mathvariant="bold">α</mi></msup><msup><mi mathvariant="bold">b</mi><mi mathvariant="bold">α</mi></msup></mfrac></math> Nếu a > 1 thì <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="normal">a</mi><mi mathvariant="normal">α</mi></msup><mo> </mo><mo>></mo><mo> </mo><msup><mi mathvariant="normal">a</mi><mi mathvariant="normal">β</mi></msup></math> khi và chỉ khi <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">α</mi><mo> </mo><mo>></mo><mo> </mo><mi mathvariant="normal">β</mi></math> Nếu a < 1 thì <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="normal">a</mi><mi mathvariant="normal">α</mi></msup><mo> </mo><mo>></mo><mo> </mo><msup><mi mathvariant="normal">a</mi><mi mathvariant="normal">β</mi></msup></math> khi và chỉ khi <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">α</mi><mo> </mo><mo><</mo><mo> </mo><mi mathvariant="normal">β</mi></math>