平方根と立方根

平方根・立方根の計算

① $a \gt 0$のとき

$\quad \quad (\pm \sqrt{a})^2 = a$

② $\sqrt{a^2} = |a| = \begin{cases} & a \quad (a\geq 0 のとき) \\ – & a \quad (a\lt 0 のとき) \end{cases}$

③ $a \gt 0, \quad b \gt 0$のとき

$\quad \quad \sqrt{a} \sqrt{b} = \sqrt{ab}, \quad \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$

④ $a \gt 0 , \quad m \gt 0 のとき \quad \sqrt{m^2 a} = m \sqrt{a}$

⑤ $\sqrt[3]{a}\cdot \sqrt[3]{b} = \sqrt[3]{ab}, \quad \frac{\sqrt[3]{a}}{\sqrt[3]{b}} = \sqrt[3]{\frac{a}{b}}$

分母の有理化

① 平方根 $( a \gt 0, \quad b \gt 0)$

$\begin{align} \frac{b}{\sqrt{a}} &= \frac{b\cdot \sqrt{a}}{\sqrt{a} \cdot \sqrt{a}} \\ &= \frac{b \sqrt{a}}{a} \end{align}$

$\begin{align} \frac{c}{\sqrt{a}+\sqrt{b}} &= \frac{c\cdot(\sqrt{a} – \sqrt{b})}{(\sqrt{a} + \sqrt{b})\cdot (\sqrt{a} – \sqrt{b})} \\ &= \frac{c\cdot(\sqrt{a} – \sqrt{b})}{a – b} \end{align}$

② 立方根

$\begin{align} \frac{b}{\sqrt[3]{a}} &= \frac{b\cdot \sqrt[3]{a^2}}{\sqrt[3]{a} \cdot \sqrt[3]{a^2}} \\ &= \frac{b \sqrt[3]{a^2}}{a} \end{align}$

$\begin{align} \frac{c}{\sqrt[3]{a}+\sqrt[3]{b}} &= \frac{c \cdot (\sqrt[3]{a^2} -\sqrt[3]{ab}+ \sqrt[3]{b^2})}{ (\sqrt[3]{a} + \sqrt[3]{b}) \cdot (\sqrt[3]{a^2} -\sqrt[3]{ab}+ \sqrt[3]{b^2}) } \\ &= \frac{c \cdot ( \sqrt[3]{a^2} -\sqrt[3]{ab}+ \sqrt[3]{b^2} ) }{a + b} \end{align}$

※$(x+y)(x^2-xy+y^2) = x^3+y^3$の公式を利用。

二重根号

$x \gt 0 , \quad y \gt 0$のとき

$\sqrt{x+y+2\sqrt{xy}}=\sqrt{(\sqrt{x}+\sqrt{y})^2} = |\sqrt{x}+\sqrt{y}| = \sqrt{x} + \sqrt{y}$

$\sqrt{x+y-2\sqrt{xy}}=\sqrt{(\sqrt{x}-\sqrt{y})^2} = |\sqrt{x}-\sqrt{y}|$