Решебник по алгебре 8 класс Макарычев ФГОС Задание 477

 

\[\boxed{\text{477\ (477).}\text{\ }\text{Еуроки\ -\ ДЗ\ без\ мороки}}\]

\[Воспользуемся\ формулой\ \]

\[разности\ квадратов:\]

\[\mathbf{a}^{\mathbf{2}}\mathbf{-}\mathbf{b}^{\mathbf{2}}\mathbf{=}\left( \mathbf{a - b} \right)\left( \mathbf{a + b} \right)\mathbf{.}\]

Решение.

\[\textbf{а)}\ \sqrt{\frac{165^{2} - 124^{2}}{164}} =\]

\[= \sqrt{\frac{(165 - 124)(165 + 124)}{164}} =\]

\[= \sqrt{\frac{41 \cdot 289}{41 \cdot 4}} = \frac{\sqrt{289}}{\sqrt{4}} = \frac{17}{2} = 8,5\]

\[\textbf{б)}\ \sqrt{\frac{98}{176^{2} - 112^{2}}} =\]

\[= \sqrt{\frac{98}{(176 - 112)(176 + 112)}} =\]

\[= \sqrt{\frac{49 \cdot 2}{64 \cdot 288}} = \sqrt{\frac{49}{64 \cdot 144}} =\]

\[= \frac{\sqrt{49}}{\sqrt{64} \cdot \sqrt{144}} = \frac{7}{8 \cdot 12} = \frac{7}{96}\]

\[\textbf{в)}\ \sqrt{\frac{149^{2} - 76^{2}}{457^{2} - 384^{2}}} =\]

\[= \sqrt{\frac{(149 - 76)(149 + 76)}{(547 - 384)(457 + 384)}} =\]

\[= \sqrt{\frac{73 \cdot 225}{73 \cdot 841}} = \frac{\sqrt{225}}{\sqrt{841}} = \frac{15}{29}\]

\[\textbf{г)}\ \sqrt{\frac{{145,5}^{2} - {96,5}^{2}}{{193,5}^{2} - {31,5}^{2}}} =\]

\[= \sqrt{\frac{(145,5 - 96,5)(145,5 + 96,5)}{(193,5 - 31,5)(193,5 + 31,5)}} =\]

\[= \sqrt{\frac{49 \cdot 242}{162 \cdot 225}} =\]

\[= \sqrt{\frac{49 \cdot 121}{81 \cdot 225}} = \frac{\sqrt{49} \cdot \sqrt{121}}{\sqrt{81} \cdot \sqrt{225}} =\]

\[= \frac{7 \cdot 11}{9 \cdot 15} = \frac{77}{135}\]

\[\boxed{\text{477.}\text{\ }\text{еуроки}\text{-}\text{ответы}\text{\ }\text{на}\text{\ }\text{пятёрку}}\]

Пояснение.

Решение.

\[\textbf{а)}\ \sqrt{y^{4}} = y^{2}\]

\[\sqrt{y^{4}} = \sqrt{\left( y^{2} \right)^{2}} = \left| y^{2} \right|\text{\ \ }\]

\[y^{2} \geq 0 - при\ любом\ y;\]

\[верно\ при\ y - любое\ число.\]

\[\textbf{б)}\ \sqrt{x^{12}} = x^{6}\]

\[\sqrt{x^{12}} = \sqrt{\left( x^{6} \right)^{2}} = \left| x^{6} \right|\]

\[x^{6} \geq 0 - при\ любом\ x;\]

\[верно\ при\ x - любое\ число.\]

\[\textbf{в)}\ \sqrt{x^{6}} = x^{3}\]

\[\sqrt{x^{6}} = \sqrt{\left( x^{3} \right)^{2}} = \left| x^{3} \right|\]

\[x^{3} \geq 0\ \ при\ x \geq 0;\]

\[верно\ при\ x \geq 0.\]

\[\textbf{г)}\ \sqrt{c^{10}} = - c^{5}\]

\[\sqrt{c^{10}} = \sqrt{\left( c^{5} \right)^{2}} = \left| c^{5} \right|\]

\[c^{5} \leq 0\ \ при\ c \leq 0;\]

\[верно\ при\ c \leq 0.\]

\[\textbf{д)}\ \sqrt{a^{14}} = - a^{7}\]

\[\sqrt{a^{14}} = \sqrt{\left( a^{7} \right)^{2}} = \left| a^{7} \right|\]

\[a^{7} \leq 0\ \ при\ a \leq 0;\]

\[верно\ при\ a \leq 0.\ \]

\[\textbf{е)}\ \sqrt{b^{8}} = b^{4}\text{\ \ }\]

\[\sqrt{b^{8}} = \sqrt{\left( b^{4} \right)^{2}} = \left| b^{4} \right|\]

\[b^{4} \geq 0\ \ при\ любом\ значении\ b;\]

\[верно\ при\ b - любое\ число.\]