вправа 5.23 гдз 10 клас математика Мерзляк Номіровський 2018
Вправа 5.23
Умова:
Умова:
Скоротіть дріб.
Відповідь ГДЗ:
\begin{equation}
1)\frac{\sqrt{a}-\sqrt{b}}{\sqrt[4]{a}+\sqrt[4]{b}}=
\end{equation}
\begin{equation}
=\frac{ \left ( \sqrt[4]{a} \right )^{2}- \left ( \sqrt[4]{b} \right )^{2}}{\sqrt[4]{a}+\sqrt[4]{b}}=
\end{equation}
\begin{equation}
=\frac{ \left ( \sqrt[4]{a}-\sqrt[4]{b} \right ) \left ( \sqrt[4]{a}+\sqrt[4]{b} \right )}{\sqrt[4]{a}+\sqrt[4]{b}}=
\end{equation}
\begin{equation}
=\sqrt[4]{a}-\sqrt[4]{b},
\end{equation}
\begin{equation}
a > 0, b > 0;
\end{equation}
\begin{equation}
2)\frac{\sqrt[6]{x}-9}{\sqrt[12]{x}+3}=
\end{equation}
\begin{equation}
=\frac{\left ( \sqrt[12]{x}-3 \right ) \left ( \sqrt[12]{x}+3 \right )}{\sqrt[12]{x}+3}=
\end{equation}
\begin{equation}
=\sqrt[12]{x}-3,
\end{equation}
\begin{equation}
x \geq 0;
\end{equation}
\begin{equation}
3)\frac{\sqrt[8]{ab^{2}}-\sqrt[8]{a^{2}b}}{\sqrt[4]{a}-\sqrt[4]{b}}=
\end{equation}
\begin{equation}
=\frac{\sqrt[8]{ab} \left ( \sqrt[8]{b^{2}}-\sqrt[8]{a^{2}} \right )}{\sqrt[4]{a}-\sqrt[4]{b}}=
\end{equation}
\begin{equation}
=\frac{\sqrt[8]{ab} \left ( \sqrt[4]{b}-\sqrt[4]{a} \right )}{\sqrt[4]{a}-\sqrt[4]{b}}=
\end{equation}
\begin{equation}
=-\sqrt[8]{ab^{2}},
\end{equation}
\begin{equation}
a > 0, b > 0; a \neq b;
\end{equation}
\begin{equation}
4)\frac{\sqrt[3]{x^{2}}+4\sqrt[8]{x}+16}{x-64}=
\end{equation}
\begin{equation}
=\frac{\sqrt[3]{x^{2}}+4\sqrt[8]{x}+16}{\left ( \sqrt[3]{x}-4 \right ) \left ( \sqrt[3]{x^{2}}+4\sqrt[3]{x}+16 \right )} =
\end{equation}
\begin{equation}
=\frac{1}{\sqrt[3]{x}-4},
\end{equation}
\begin{equation}
x \neq 64;
\end{equation}
\begin{equation}
5)\frac{\sqrt{a}+\sqrt{b}}{\sqrt[3]{a}-\sqrt[6]{ab}+\sqrt[3]{b}}=
\end{equation}
\begin{equation}
= \frac{\left ( \sqrt[6]{a}+\sqrt[6]{b} \right ) \left ( \sqrt[3]{a}-\sqrt[6]{ab}+\sqrt[3]{b} \right )}{\sqrt[3]{a}-\sqrt[6]{ab}+\sqrt[3]{b}}=
\end{equation}
\begin{equation}
=\sqrt[6]{a}+\sqrt[6]{b},
\end{equation}
\begin{equation}
a > 0, b > 0;
\end{equation}
\begin{equation}
6) \frac{\sqrt[4]{a^{3}}-\sqrt[4]{a}+\sqrt{a}-1}{a-\sqrt{a}}=
\end{equation}
\begin{equation}
=\frac{\sqrt[4]{a} \left ( \sqrt{a}-1 \right ) + \left ( \sqrt{a}-1 \right )}{\sqrt{a} \left ( \sqrt{a}-1 \right )}=
\end{equation}
\begin{equation}
=\frac{ \left ( \sqrt{a}-1 \right ) \left ( \sqrt[4]{a}+1 \right )}{\sqrt{a} \left ( \sqrt{a}-1 \right )}=
\end{equation}
\begin{equation}
=\frac{\sqrt[4]{a}+1}{\sqrt{a}},
\end{equation}
\begin{equation}
a > 0, a \neq 1.
\end{equation}
