Числовые степенные выражения (страница 4)

Так как \(2^{3,5} = 2^3\cdot 2^{0,5} = 8\cdot\sqrt{2} = 2\cdot 4\cdot\sqrt{2}\), то

\(\sqrt{2^4 + 2^1 - 2^{3,5}} + \sqrt{2} = \sqrt{4^2 - 2\cdot 4 \cdot \sqrt{2} + 2} + \sqrt{2} = \sqrt{4^2 - 2\cdot 4 \cdot \sqrt{2} + (\sqrt{2})^2} + \sqrt{2} =\)
\(= \sqrt{(4 - \sqrt{2})^2} + \sqrt{2} = |4 - \sqrt{2}| + \sqrt{2} = 4 - \sqrt{2} + \sqrt{2} = 4\)

(так как \(4 > 2 > \sqrt{2}\), то \(4 - \sqrt{2} > 0\) и \(|4 - \sqrt{2}| = 4 - \sqrt{2}\)).

Ответ: 4

\(\sqrt{4^{\sqrt{3}} + 2^{\sqrt{3} + 1} + 1} - 2^{\sqrt{3}} = \sqrt{(2^{\sqrt{3}})^2 + 2\cdot 2^{\sqrt{3}} \cdot 1 + 1} - 2^{\sqrt{3}} = \sqrt{(2^{\sqrt{3}} + 1)^2} - 2^{\sqrt{3}} =|2^{\sqrt{3}} + 1| - 2^{\sqrt{3}} = 2^{\sqrt{3}} + 1 - 2^{\sqrt{3}} = 1\)

(так как \(2^{\sqrt{3}} + 1 > 0\), то \(|2^{\sqrt{3}} + 1| = 2^{\sqrt{3}} + 1\)).

Ответ: 1

\[12^{\sqrt[12]{\sqrt{2}}} \cdot 12^{3 - \sqrt[12]{\sqrt[3]{2}}\cdot \sqrt[6]{\sqrt[12]{2}}} = 12^{\sqrt[12]{\sqrt{2}}} \cdot 12^{3 - \sqrt[12]{\sqrt[3]{2}}\cdot \sqrt[12]{\sqrt[6]{2}}} = 12^{\sqrt[12]{\sqrt{2}}} \cdot 12^{3 - \sqrt[12]{\sqrt{2}}} = 12^{\sqrt[12]{\sqrt{2}} + 3 - \sqrt[12]{\sqrt{2}}} = 12^3 = 1728.\]

Ответ: 1728

Так как \(3^n = ((\sqrt{3})^2)^n = ((\sqrt{3})^n)^2\), то

\(\sqrt{3^{\sqrt{2}} - 2\cdot \sqrt{3}^{\sqrt{2}} + 1} - \sqrt{3}^{\sqrt{2}} = \sqrt{\left(\sqrt{3}^{\sqrt{2}}\right)^2 - 2\cdot \sqrt{3}^{\sqrt{2}}\cdot 1 + 1} - \sqrt{3}^{\sqrt{2}} =\)

\(=\sqrt{\left(\left(\sqrt{3}^{\sqrt{2}}\right) - 1\right)^2} - \sqrt{3}^{\sqrt{2}} = |\sqrt{3}^{\sqrt{2}} - 1| - \sqrt{3}^{\sqrt{2}} = \sqrt{3}^{\sqrt{2}} - 1 - \sqrt{3}^{\sqrt{2}} = -1\)

(так как \(\sqrt{3}^{\sqrt{2}} > \sqrt{3} > 1\),то \(\sqrt{3}^{\sqrt{2}} - 1 > 0\) и \(|\sqrt{3}^{\sqrt{2}} - 1| = \sqrt{3}^{\sqrt{2}} - 1\)).

Ответ: -1

\(\dfrac{1}{2^{50}}:\left(\dfrac{1}{2}\cdot\dfrac{1}{2^2}\cdot\dfrac{1}{2^3}\cdot\dfrac{1}{2^4}\cdot ...\cdot \dfrac{1}{2^{10}}\right) = \dfrac{1}{2^{50}}:\dfrac{1}{2\cdot 2^2\cdot 2^3\cdot 2^4\cdot ...\cdot 2^{10}} = \dfrac{1}{2^{50}}:\dfrac{1}{2^{1 + 2 + 3 + 4 + ... + 10}} = \dfrac{1}{2^{50}}:\dfrac{1}{2^{55}} = \dfrac{2^{55}}{2^{50}} = 2^5 = 32\).

Ответ: 32

\(\dfrac{1}{2^{101}}:\left(\dfrac{1}{2^1}\cdot\dfrac{1}{2^3}\cdot\dfrac{1}{2^5}\cdot\dfrac{1}{2^7}\cdot ...\cdot \dfrac{1}{2^{19}}\right) = \dfrac{1}{2^{101}}:\left(\dfrac{1}{2^{1 + 3 + 5 + 7 + ... + 19}}\right) =\)
\( = \dfrac{1}{2^{101}}:\left(\dfrac{1}{2^{1 + 2 + 3 + 4 + 5 + ... + 19 - (2 + 4 + 6 + ... + 18)}}\right) = \dfrac{1}{2^{101}}:\left(\dfrac{1}{2^{1 + 2 + 3 + 4 + 5 + ... + 19 - 2\cdot(1 + 2 + 3 + ... + 9)}}\right) = \dfrac{1}{2^{101}}:\left(\dfrac{1}{2^{190 - 2\cdot 45}}\right) = \dfrac{1}{2^{101}}:\dfrac{1}{2^{100}} = \dfrac{2^{100}}{2^{101}} = 2^{100 - 101} = 2^{-1} = 0,5\).

Ответ: 0,5