Bài 19 trang 82 SGK Đại số và Giải tích 12 Nâng cao

Đơn giản biểu thức

LG a

$a^{-2\sqrt{2}}{\left(\frac{1}{a^{-\sqrt{2}-1}}\right)}^{\sqrt{2}+1}$; 

Lời giải chi tiết:

$a^{-2\sqrt{2}}{\left(\frac{1}{a^{-\sqrt{2}-1}}\right)}^{\sqrt{2}+1}$

$=a^{-2\sqrt{2}}.{\left[{\left(a^{-\sqrt{2}-1}\right)}^{-1}\right]}^{\sqrt{2}+1}$

$=a^{-2\sqrt{2}}{\left(a^{\sqrt{2}+1}\right)}^{\sqrt{2}+1}$

$=a^{-2\sqrt{2}}.a^{\left(\sqrt{2}+1\right)\left(\sqrt{2}+1\right)}$

$=a^{-2\sqrt{2}}{a}^{3+2\sqrt{2}}=a^{-2\sqrt{2}+3+2\sqrt{2}}$

$=a^3$

LG b

${\left(\frac{a^{\sqrt{3}}}{b^{\sqrt{3}-1}}\right)}^{\sqrt{3}+1}\frac{a^{-1-\sqrt{3}}}{b^{-2}};$

Lời giải chi tiết:

${\left(\frac{a^{\sqrt{3}}}{b^{\sqrt{3}-1}}\right)}^{\sqrt{3}+1}\frac{a^{-1-\sqrt{3}}}{b^{-2}}$

$=\frac{{\left(a^{\sqrt{3}}\right)}^{\sqrt{3}+1}}{{\left(b^{\sqrt{3}-1}\right)}^{\sqrt{3}+1}}.\frac{a^{-1-\sqrt{3}}}{b^{-2}}$

$=\frac{a^{\sqrt{3}.\left(\sqrt{3}+1\right)}}{b^{\left(\sqrt{3}-1\right)\left(\sqrt{3}+1\right)}}.\frac{a^{-1-\sqrt{3}}}{b^{-2}}$

$=\frac{a^{3+\sqrt{3}}}{b^2}.\frac{a^{-1-\sqrt{3}}}{b^{-2}}$

$=\frac{a^{3+\sqrt{3}}.a^{-1-\sqrt{3}}}{b^2.b^{-2}}=\frac{a^{3+\sqrt{3}-1-\sqrt{3}}}{b^{2-2}}=\frac{a^2}{b^0}=a^2$

LG c

$\frac{a^{2\sqrt{2}}-b^{2\sqrt{3}}}{{\left(a^{\sqrt{2}}-b^{\sqrt{3}}\right)}^2}+1;$ 

Lời giải chi tiết:

$\frac{a^{2\sqrt{2}}-b^{2\sqrt{3}}}{{\left(a^{\sqrt{2}}-b^{\sqrt{3}}\right)}^2}+1=\frac{a^{2\sqrt{2}}-b^{2\sqrt{3}}+{\left(a^{\sqrt{2}}-b^{\sqrt{3}}\right)}^2}{{\left(a^{\sqrt{2}}-b^{\sqrt{3}}\right)}^2}$

$=\frac{a^{2\sqrt{2}}-b^{2\sqrt{3}}+a^{2\sqrt{2}}-2a^{\sqrt{2}}.b^{\sqrt{3}}+b^{2\sqrt{3}}}{{\left(a^{\sqrt{2}}-b^{\sqrt{3}}\right)}^2}$

$=\frac{2a^{2\sqrt{2}}-2a^{\sqrt{2}}{b}^{\sqrt{3}}}{{\left(a^{\sqrt{2}}-b^{\sqrt{3}}\right)}^2}$

$=\frac{2a^{\sqrt{2}}\left(a^{\sqrt{2}}-b^{\sqrt{3}}\right)}{{\left(a^{\sqrt{2}}-b^{\sqrt{3}}\right)}^2}$

$=\frac{2a^{\sqrt{2}}}{a^{\sqrt{2}}-b^{\sqrt{3}}}$

LG d

$\sqrt{{\left(x^{\pi }+y^{\pi }\right)}^2-{\left(4^{\frac{1}{\pi }}xy\right)}^{\pi }};$

Lời giải chi tiết:

$\sqrt{{\left(x^{\pi }+y^{\pi }\right)}^2-{\left(4^{\frac{1}{\pi }}xy\right)}^{\pi }}$

$\begin{array}{l}=\sqrt{{\left(x^{\pi }\right)}^2+2x^{\pi }{y}^{\pi }+{\left(y^{\pi }\right)}^2-{\left(4^{\frac{1}{\pi }}\right)}^{\pi }{x}^{\pi }{y}^{\pi }}\\ =\sqrt{x^{2\pi }+2x^{\pi }{y}^{\pi }+y^{2\pi }-4x^{\pi }{y}^{\pi }}\end{array}$

$=\sqrt{x^{2\pi }+y^{2\pi }-2x^{\pi }{y}^{\pi }}$

$=\sqrt{{\left(x^{\pi }-y^{\pi }\right)}^2}$

$=\left|x^{\pi }-y^{\pi }\right|$.

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