notes/notes/mat/Mocniny.md

tags
mat

Mocniny

a+a+a+a+a = 5a násobení a*a*a*a*a = a^5 mocnina a = základ mocniny ^5 = exponent

1. n^0 = 1
2. a^r+a^n nelze udělat nic
3. a^r*a^n=a^{r+n}
4. 4. \frac{a^s}{a^r}={a^{s-r}}
5. \frac{a^3*a^{-2}}{(a^2)^3}=\frac{a^{2-3}}{a^{2*3}}=\frac{a^1}{a^6}=a^{1-6}=a^5
6. (\frac{a}{b})^2=\frac{a^2}{b^2}
7. \forall a,b \in R (a*b)^r = a^r * b^r
8. \forall a,b \in R \wedge b \neq 0 (\frac{a}{b})^r=\frac{a^r}{b^r} (r \in Q)

\sqrt[y]{x} = x^\frac{1}{y}

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\sqrt3 * 3 ^ 2 * (\sqrt[3]{3})^3 = \sqrt3 * 9 * 3 = 27 \sqrt3

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$$ \sqrt[3]{7} * 49 * 7 \sqrt7 = 7^\frac{1}{3} * 7^2 * 7^1 * 7^\frac{1}{2}

$$ \frac{\sqrt{a} * \sqrt[3]b}{a^\frac{-1}{3} * b^2} * \frac{a^{-3}}{b^\frac{1}{4}} = \frac{a^\frac{1}{2} * b^\frac{1}{3}}{a^{-1}{2} * b^\frac{2}{1}} * \frac{a^{-3}{1}}{b^\frac{1}{4}}

$$ \frac{225^3 * 243}{15^2 * 45^3} = \frac{15 ^ 6 * 243}{5^2 * 3^2 * 3^2 * 15^2} = \frac{5^6 * 3^11}{5^4 * 3^6} = 5^2 * 3^5

$$ \frac{\sqrt[4]{\sqrt{a}\sqrt[3]{a^2}}}{a^\frac{5}{3}} = \frac{(a^\frac{1}{2}*a^\frac{2}{3})^\frac{1}{4}}{a^\frac{5}{3}} = \frac{(a^\frac{7}{6})^\frac{1}{4}}{a^\frac{5}{3}} = \frac{a^\frac{7}{24}}{a^\frac{5}{3}} = a^\frac{7}{24} * a^\frac{3}{5} = a^\frac{7}{40} $$ $$ \frac{1}{2} + \frac{2}{3} = \frac{3}{6} + \frac{4}{6} = \frac{7}{6}

$$ (x^4 + 2x^2 - 3x + 5) + (3x^3 - 2x^2 + x - 4) = x^4 + 3x^3 + 2x^2 - 2x^2 -3x + x + 5 - 4 = x^4 + 3x^3 - 2x + 1

$$ (3x^2 + 21x + 24) : (x - 1) = 3x + 24

$$ (6x^2 + 8x^5 + 14x^4 - 21x^3 + 3x^2 + 8x - 14) : (3x^3 - x^2 + 2)