ГДЗ по алгебре 8 класс Макарычев ФГОС Задание 165

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Пояснение.

Решение.

\[\textbf{а)}\ \left( n + \frac{1}{n} \right)^{2} =\]

\[= n^{2} + 2n \cdot \frac{1}{n} + \frac{1}{n^{2}} =\]

\[= n^{2} + 2 + \frac{1}{n^{2}} =\]

\[= \frac{n^{4} + 2n^{2} + 1}{n²}\]

\[\textbf{б)}\ \left( \frac{a^{\backslash a}}{b} - \frac{b^{\backslash b}}{a} \right)^{2} =\]

\[= \frac{a^{4} - 2a^{2}b^{2} + b^{4}}{a²b²}\]

\[\textbf{в)}\ \left( \frac{x}{y} + 1 \right)^{2} + \left( \frac{x}{y} - 1 \right)^{2} =\]

\[= \frac{x^{2}}{y^{2}} + 2\frac{x}{y} + 1 + \frac{x^{2}}{y^{2}} - 2\frac{x}{y} + 1 =\]

\[= \frac{2x^{2}}{y^{2}} + 2 = \frac{2x^{2} + 2y²}{y²}\]

\[\textbf{г)}\ \left( \frac{p}{q} + \frac{q}{p} \right)^{2} - \left( \frac{p}{q} - \frac{q}{p} \right)^{2} =\]

\[= \frac{p^{2}}{q^{2}} + \frac{2pq}{\text{pq}} + \frac{q^{2}}{p^{2}} - \frac{p^{2}}{q^{2}} + \frac{2pq}{\text{qp}} - \frac{q^{2}}{p^{2}} =\]

\[= \frac{2pq}{\text{qp}} + \frac{2pq}{\text{qp}} = \frac{4pq}{\text{pq}} = 4\]