tex]...

July 2022 1 10 Report

упростите выражения
[tex] \sqrt[3]{24} - \sqrt[3]{3} [/tex]
[tex] \sqrt[4]{512} - \sqrt[4]{2} [/tex]
[tex] \frac{ \sqrt{ab} \sqrt[4]{a} }{ (a+b) \sqrt[4]{ \frac{ b^{2} }{a} } } - \frac{ a^{2}+ b^{2} }{ a^{2} - b^{2} } [/tex]

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Artem112

Verified answer

$\sqrt[3]{24} - \sqrt[3]{3}=2 \sqrt[3]{3} - \sqrt[3]{3}=\sqrt[3]{3}
\\\
\sqrt[4]{512} - \sqrt[4]{2}=4\sqrt[4]{2} - \sqrt[4]{2}=3\sqrt[4]{2}$

$\frac{ \sqrt{ab} \sqrt[4]{a} }{ (a+b) \sqrt[4]{ \frac{ b^{2} }{a} } } - \frac{ a^{2}+ b^{2} }{ a^{2} - b^{2} } =
 \frac{ \sqrt{ab} \sqrt[4]{a}\sqrt[4]{a } }{ (a+b) \sqrt[4]{ b^{2} } } - \frac{ a^{2}+ b^{2} }{ a^{2} - b^{2} } =
 \frac{ \sqrt{ab} \sqrt{a} }{ (a+b) \sqrt{ b } } - \frac{ a^{2}+ b^{2} }{ a^{2} - b^{2} } =
\\\
 \frac{ a }{ a+b} } - \frac{ a^{2}+ b^{2} }{ a^{2} - b^{2} } =
 \frac{ a(a-b)-a^2-b^2 }{ a^{2} - b^{2} } }= \frac{ a^2-ab-a^2-b^2 }{ a^{2} - b^{2} } }= 
$
$= \frac{ -ab-b^2 }{ a^{2} - b^{2} } }=\frac{ -b(a+b) }{ (a - b)(a+b) } }=\frac{ b }{ b-a } }$

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