вправа 5.24 гдз 10 клас математика Мерзляк Номіровський 2018
Вправа 5.24
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Скоротіть дріб.
Відповідь ГДЗ:
\begin{equation} 1)\frac{\sqrt[6]{a}+1}{\sqrt[3]{a}-1}= \end{equation} \begin{equation} =\frac{\sqrt[6]{a}+1}{\left ( \sqrt[6]{a}-1 \right ) \left ( \sqrt[6]{a}+1 \right )}= \end{equation} \begin{equation} =\frac{1}{\sqrt[6]{a}-1}, \end{equation} \begin{equation} a > 0; a \neq 1; \end{equation} \begin{equation} 2) \frac{\sqrt{m}-\sqrt[4]{mn}}{\sqrt[4]{mn}-\sqrt{n}}= \end{equation} \begin{equation} =\frac{\sqrt[4]{m} \left ( \sqrt[4]{m}-\sqrt[4]{n} \right )}{\sqrt[4]{n} \left ( \sqrt[4]{m}-\sqrt[4]{n} \right )}= \end{equation} \begin{equation} =\frac{\sqrt[4]{m}}{\sqrt[4]{n}}=\sqrt[4]{\frac{m}{n}}, \end{equation} \begin{equation} m > 0, n > 0, m \neq n; \end{equation} \begin{equation} 3)\frac{a-b}{\sqrt[3]{a}-\sqrt[3]{b}}= \end{equation} \begin{equation} =\frac{ \left ( \sqrt[3]{a}-\sqrt[3]{b} \right ) \left ( \sqrt[3]{a^{2}}+\sqrt[3]{b^{2}} \right )}{\sqrt[3]{a}-\sqrt[3]{b}}= \end{equation} \begin{equation} =\sqrt[3]{a^{2}}+\sqrt[3]{ab}+\sqrt[3]{b^{2}}, \end{equation} \begin{equation} a \neq b; \end{equation} \begin{equation} 4)\frac{a\sqrt{b}-b\sqrt{a}}{\sqrt{ab}}= \end{equation} \begin{equation} =\frac{\sqrt{ab} \left ( \sqrt{a}-\sqrt{b} \right )}{\sqrt{ab}}= \end{equation} \begin{equation} =\sqrt{a}-\sqrt{b}, \end{equation} \begin{equation} a > 0, b > 0. \end{equation}
\begin{equation} 1)\frac{\sqrt[6]{a}+1}{\sqrt[3]{a}-1}= \end{equation} \begin{equation} =\frac{\sqrt[6]{a}+1}{\left ( \sqrt[6]{a}-1 \right ) \left ( \sqrt[6]{a}+1 \right )}= \end{equation} \begin{equation} =\frac{1}{\sqrt[6]{a}-1}, \end{equation} \begin{equation} a > 0; a \neq 1; \end{equation} \begin{equation} 2) \frac{\sqrt{m}-\sqrt[4]{mn}}{\sqrt[4]{mn}-\sqrt{n}}= \end{equation} \begin{equation} =\frac{\sqrt[4]{m} \left ( \sqrt[4]{m}-\sqrt[4]{n} \right )}{\sqrt[4]{n} \left ( \sqrt[4]{m}-\sqrt[4]{n} \right )}= \end{equation} \begin{equation} =\frac{\sqrt[4]{m}}{\sqrt[4]{n}}=\sqrt[4]{\frac{m}{n}}, \end{equation} \begin{equation} m > 0, n > 0, m \neq n; \end{equation} \begin{equation} 3)\frac{a-b}{\sqrt[3]{a}-\sqrt[3]{b}}= \end{equation} \begin{equation} =\frac{ \left ( \sqrt[3]{a}-\sqrt[3]{b} \right ) \left ( \sqrt[3]{a^{2}}+\sqrt[3]{b^{2}} \right )}{\sqrt[3]{a}-\sqrt[3]{b}}= \end{equation} \begin{equation} =\sqrt[3]{a^{2}}+\sqrt[3]{ab}+\sqrt[3]{b^{2}}, \end{equation} \begin{equation} a \neq b; \end{equation} \begin{equation} 4)\frac{a\sqrt{b}-b\sqrt{a}}{\sqrt{ab}}= \end{equation} \begin{equation} =\frac{\sqrt{ab} \left ( \sqrt{a}-\sqrt{b} \right )}{\sqrt{ab}}= \end{equation} \begin{equation} =\sqrt{a}-\sqrt{b}, \end{equation} \begin{equation} a > 0, b > 0. \end{equation}