Степени и корни

aⁿ =	a ⋅ a ⋅ a ⋅ ... ⋅ a ⋅ a ⋅ a
	n раз

a ∈ R, b ∈ R, b ≠ 0, m ∈ Z, n ∈ Z

Свойства степеней

1) a⁰ = 1

2) a¹ = a

3) 1ⁿ = 1

4) aⁿa^m = a^{n + m}

5) $\frac{a^{n}}{a^{m}} = a^{n - m}$

6) (ab)ⁿ = aⁿbⁿ

7) ${(\frac{a}{b})}^{n} = \frac{a^{n}}{b^{n}}$

8) (aⁿ)^m = a^nm

9) $a^{- n} = \frac{1}{a^{n}}$

10) ${(\frac{a}{b})}^{- n} = {(\frac{b}{a})}^{n}$

1) $\sqrt[n]{a^{n}} = a$

2) $a^{\frac{n}{m}} = \sqrt[m]{a^{n}}$

3) ${(\sqrt[m]{a})}^{n} = \sqrt[m]{a^{n}}$

4) $\sqrt[n]{\sqrt[m]{a}} = \sqrt[n m]{a}$

5) $\sqrt[k n]{a^{k m}} = \sqrt[m]{a^{n}}$

6) $\sqrt[n]{a b} = \sqrt[n]{a} \cdot \sqrt[n]{b}$

7) $\sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}}$

8) $\sqrt[m]{a} \sqrt[n]{b} = \sqrt[m n]{a^{n} b^{m}}$

f(x)^g₁(x) − f(x)^g₂(x) > 0 =>

(f(x) − 1)(g₁(x) − g₂(x)) > 0

(a^b − a^c ≤ 0 => (a − 1)(b − c) ≤ 0)