Question

((sqrt(2)) ^ sqrt(2)) ^ sqrt(2)

Answer 1

Ответ:

${\left(\big(\sqrt{2}\, \big) ^ \sqrt{2} \right)} ^{ \sqrt{2} }=2$

Решение:

${\left(\big(\sqrt{2}\, \big) ^ \sqrt{2} \right)} ^{ \sqrt{2} }=\left(\sqrt{2}\, \right) ^ {\sqrt{2} \cdot \sqrt{2} }=\left(\sqrt{2}\, \right) ^ 2=2$

Элементы теории:

Формула для возведения степени в степень:

$\big(a^m\big)^n=a^{mn}$

Некоторые свойства корней:

$\sqrt{a} \cdot \sqrt{a} =\big(\sqrt{a}\,\big)^2=a,\ a\geqslant 0$

Answer 2

To calculate the expression ((√2)^√2)^√2, you can follow these steps:

1. First, calculate (√2)^√2: (√2)^√2 ≈ 2^(1/2)^(1/2) ≈ 2^(1/2 * 1/2) ≈ 2^(1/4)

2. Now, calculate ((√2)^√2)^√2 using the result from step 1: ((√2)^√2)^√2 ≈ (2^(1/4))^√2

3. To simplify further, multiply the exponents: (2^(1/4))^√2 = 2^(1/4 * √2)

So, ((√2)^√2)^√2 is approximately equal to 2^(1/4 * √2). This is the simplified expression.